














































































































Copyright^?_13&S? 


COPYRIGHT DEPOSm 















HIGH SCHOOL 
TRIGONOMETRY 


BY 

DAVID RAYMOND CURTISS 

AND 

ELTON JAMES MOULTON 

PROFESSORS OF MATHEMATICS, NORTHWESTERN 
UNIVERSITY 


WITH TABLES 


D. C. HEATH AND COMPANY 


BOSTON 

ATLANTA 


NEW YORK 
SAN FRANCISCO 
LONDON 


CHICAGO 

DALLAS 


X. Q» 


Q A 53 \ 


Copyright, 1927 and 1928 
By D. C. Heath and Company 

2 c 8 



PRINTED IN U.S.A. 



-7 1928 

C1A1069647 


>V). u . q . 


PREFACE 


In preparing a text on Plane Trigonometry adapted to the 
needs of high schools, the authors have had especially in mind 
classes reciting from three to five times a week for a half year. 
The material is so presented as to make it easy to lay out 
courses of varying length. 

A distinguishing feature of this book is its fulness of ex¬ 
planation. The majority of texts, prepared primarily for 
college classes, have been so brief that the instructor has had 
to supply many details of proof and practically all illustrative 
material. Such abbreviated treatments have been designed 
to answer the needs of courses where less than a full semester 
is given to trigonometry, but the expedient of cutting out 
explanation in order to shorten a course is a doubtful one. 
Especially in high school classes, the instructor can better 
employ the recitation period in other ways than in supple¬ 
menting the text. The authors of the present volume have 
therefore included an ample amount of explanatory material 
including many illustrative exercises. They have, however, 
endeavored to avoid diffuseness and the inclusion of unneces¬ 
sary detail. 

If starred sections are omitted the text can easily be covered 
in from 50 to 60 recitation periods. Classes meeting daily can 
include the starred sections. In assigning lessons an instructor 
who has used briefer texts should bear in mind that, on ac¬ 
count of the greater amount of explanatory material in this 
book, five or six pages here often correspond to two >r three 
in the hundred page style of presentation. 

iii 



IV 


PREFACE 


Briefer courses. A survey course of fifteen lessons, 
including the solution of right and of oblique triangles 
by natural functions, is afforded by the first three 
chapters. 

In a course of thirty lessons including the theory of loga¬ 
rithms and their use in the solution of triangles, the time may 
be divided as follows: Chapters I and II, eight lessons; 
Chapters IV, V, VI, ten or twelve lessons; Chapters VIII 
and IX (with the first four sections of Chapter III), ten or 
twelve lessons. 

Another course of thirty lessons which includes all pre¬ 
requisites for analytic geometry and the calculus, but does 
not use logarithms, would cover nearly all the ground of the 
first six or seven chapters, omitting starred sections except 
in Chapter III. Of these thirty lessons from twelve to fifteen 
should be devoted to the first three chapters. The omission 
of computation by means of logarithms, as contemplated in 
this program, would be in accord with the growing tendency 
to calculate with slide rules, machines, and multiplication 
tables. 

Early use of coordinates and the general angle. Instead 
of beginning with acute angles, the definitions of the first 
chapter apply to angles of any magnitude. This saves time 
and in the end proves less confusing to the student. Co- 
Ordinates, both rectangular and polar, are used in these 
definitions for three reasons. The first is that the problem 
of locating a position by its coordinates is a practical one 
giving a natural approach to the consideration of the trigo¬ 
nometric functions. In the second place the use of co¬ 
ordinates distinctly simplifies and clarifies the definitions of 
the functions. Finally such a treatment tends to unify 
trigonometry with algebra and analytic geometry. 

Generality of proofs. Proofs are given so as to apply to all 
cases. ‘ The formulas for sin (a + /3) and cos (a + /3) are 
proved first for the simplest cases, and in a later starred 


PREFACE 


V 


section it is pointed out that the same proof, properly under¬ 
stood, is of universal application. 

Tables. In Chapter II the use of four-place tables of 
squares and of natural functions is explained. Chapter III 
makes further use of these tables. In Chapter VIII there 
is an unusually full explanation of logarithms and of com¬ 
putation with both four and five place tables. If it is desired 
to use, for example, only four place tables, much explanatory 
material (chiefly examples worked out in full) relating to 
five place computation may be omitted. 

Illustrative examples. This book contains far more illus¬ 
trative examples, worked out in part or in full, than do the 
briefer texts. Almost all important topics are here repre¬ 
sented. Such examples, judiciously chosen, with enough 
detail but not too much, often impart more valuable in¬ 
struction than does a discussion confined to generalities. 

Exercises. It is hoped that the exercises are sufficiently 
numerous for longer as well as for shorter courses. They 
have been chosen with care, and are roughly graded accord¬ 
ing to difficulty so that the harder ones are toward the end of 
each set. In general, when an exercise is subdivided this has 
been done so that it will be natural to give the whole exercise 
as part of a lesson and not one subdivision of one exercise, 
another of a second, and so on. 

In the last two chapters there are sets of exercises in 
which it is required that four place tables be used, and other 
sets for five place tables. This separation of material makes 
it easy to use either sort of tables, or both kinds. The authors 
have tried to be explicit and clear in their statements, so 
that the student may know just what is required in each 
exercise. 

Significant figures. Chapter II contains a brief discussion 
of the question of the number of significant figures that 
should be retained in computation. This may be omitted in a 
brief course, though the matter is one of much practical im- 


VI 


PREFACE 


portance. The number of figures to be retained is indicated 
in exercises on applications. 

The authors wish here to express their obligation to Mr. 
M. J. Newell of the Evanston (Ill.) High School for helpful 
suggestions and advice. 


CONTENTS 


CHAPTER I 

THE SIX TRIGONOMETRIC FUNCTIONS 

SECTION PAGE 

1. Angles in plane geometry. 1 

2. Angles generated by a rotating ray. 1 

3. The general angle in trigonometry. 2 

4. Measurement of angles. 2 

5. The protractor. 3 

6. Directions measured from a line of reference. 6 

7. Location of a point by distance and direction. 6 

8. Polar coordinates. 7 

9. Directed lines. 9 

10. Projection. 9 

11. Location of a point by rectangular coordinates. 10 

12. Quadrants. .. 11 

13. Changing from rectangular to polar coordinates. 13 

14. Changing from polar to rectangular coordinates. 15 

15. The six trigonometric functions. 17 

16. Algebraic signs of the functions. 18 

17. Values of the functions by measurement. 21 

18. Applications. 22 

19. Functions of 45°, 135°, 225° and 315°. 24 

20. Functions of 30°, 60° and 120°. 25 

21. Functions of 0°, 180° and 90°. 27 

22. Problems in which a function is given. 29 

★ 23. Projections on coordinate axes. 31 

★ 24. Vectors. Components. Resultants. 32 

CHAPTER II 
RIGHT TRIANGLES 

25. The problem of solving a triangle. 35 

26. Functions of an acute angle of a right triangle. 35 

27. Functions of complementary angles. 36 

vii 





























Vlll 


CONTENTS 


SECTION 

28. Tables of values of functions of acute angles. 

PAGE 

. 37 

. 39 

30. 

Ol 

Solution of typical right triangles. 

. 41 

. 43 

o 1. 

oo 


. 44 

OA. 

QQ 


. 45 

oo. 

★ 34. 

OK 

Approximations. Significant figures. 

. 46 

. 51 

OO. 

oa 


. 52 

oo. 

37. 

Applications to heights and distances. 

. 52 


CHAPTER III 


OBLIQUE TRIANGLES 


38. General statement. 

39. Sine and cosine of obtuse angles. 

40. The law of sines. 

41. The law of cosines. 

★ 42. Another cosine formula. 

★ 43. Case I. Given two angles and one side. 

★ 44. Case II. Given two sides and the angle opposite one of 

them. 

★ 45. Case III. Given two sides and the included angle. 

★ 46. Case IV. Given three sides. 


57 

58 

59 

60 

63 

64 

66 

71 

74 


CHAPTER IV 


REDUCTION FORMULAS. LINE VALUES. GRAPHS 

47. Functions of 180° — 9 . 

48. Functions of 180° + 9 . 

49. Functions of 360° — 9 and of —9 . 

50. General rule for n • 180° =b 9 . 

51. Functions of 90° =L 9 . 

52. Functions of 270° =b 9 . 

53. General rule for n • 90° dz 9, where n is odd. 

54. Line values. 

55. Variation of sin 9 and tan 9 . 

★ 56. Graphs in rectangular coordinates. 

★ 57. Graphs of the trigonometric functions. 


79 

81 

82 

83 

86 

88 

89 

90 
92 
94 
96 
































CONTENTS 


IX 


CHAPTER V 

FUNDAMENTAL IDENTITIES 

SECTION PAGE 

58. Trigonometric identities.. 99 

59. Formulas involving one angle. 99 

★ 60. Formulas expressing the functions in terms of a single 

function. 101 

★ 61. Simplification of expressions involving trigonometric 

functions. 103 

62. Proofs of identities. 104 

63. Addition formulas.. 107 

64. Formulas for sin ( a + P) and cos (a + p) . 108 

★ 65. Cases where a and p are nolrboth between 0° and 90°.... 110 

66. Formulas for sin (ct — p) and cos (a — p) . 112 

67. Formulas for tan (a + p) and tan (a — p) . 114 

68. Formulas for the double angle. 116 

69. Formulas for the half-angle. 117 

70. Products which are equal to sums or differences of two 

sines or cosines. 121 

CHAPTER VI 

RADIAN MEASURE. INVERSE FUNCTIONS 

71. The radian. 126 

72. Relations between radians and degrees. 127 

73. Length of circular arc. 129 

★ 74. Areas of segment and sector of a circle. 132 

★ 75. Velocity of a point moving in a circle. 133 

76. Inverse trigonometric functions. Principal values. 135 

77. Determination of all values of an inverse trigonometric 

function. 137 

★ 78. Graphs of inverse functions. 140 

★ 79. Identities involving inverse functions. 141 

CHAPTER VII 

TRIGONOMETRIC EQUATIONS 

80. Definitions. 144 

81. Simple examples. 145 

82. Factorable equations. 146 


























X 


CONTENTS 


SECTION PAGE 

83. Equations reducible to quadratic form. 147 

★ 84. The type a sin x + b cos x = c ... 150 

^ 85. Approximate solutions. 151 

CHAPTER VIII 
LOGARITHMS 

86. Exponents. 154 

87. Expressing numbers as powers of 10. 156 

88. Definition of the logarithm of a number. 157 

89. Fundamental laws of logarithms. 159 

90. Characteristic and mantissa. 162 

91. Finding logarithms from a table. 164 

92. Finding a number whose logarithm is given. 167 

93. Products and quotients found by use of logarithms. 169 

★ 94. Cologarithms. 170 

95. Powers and roots. 171 

★ 96. Computations involving negative numbers. 172 

97. Logarithms of trigonometric functions. 174 

^ 98. Angles near 0° or 90°. 177 

★ 99. Change of base of logarithms. 181 

★ 100. The logarithmic scale. 182 

★ 101. The slide rule. 183 

CHAPTER IX 

SOLUTION OF TRIANGLES BY LOGARITHMS 

102. Solution of right triangles. 185 

103. The law of tangents. 189 

104. Solving oblique triangles by logarithms. 190 

105. Case I. Given two angles and one side. 191 

106. Case II. Given two sides and an angle opposite to one. 193 

107. Case III. Given two sides and the included angle .... 198 

^108. The half-angle formulas. First proof. 200 

109. The half-angle formulas. Second proof . 201 

110. Case IV. Given three sides. 203 

111. Area of a triangle. 205 

^T12. Radii of inscribed and circumscribed circles. 205 

113. Applications. 207 

Formulas. 214 
































TRIGONOMETRY 


CHAPTER I 

THE SIX TRIGONOMETRIC FUNCTIONS 

In this chapter we shall give definitions of certain expres¬ 
sions called the trigonometric functions which are of constant 
use in trigonometry. We lead up to these definitions by a 
description of several ways of locating the positions of 
objects in a plane, and by discussion of certain related 
problems. Following the definitions we consider a num¬ 
ber of special examples. The principal applications of 
the trigonometric functions will be given in succeeding 
chapters. 

1. Angles in plane geometry. The reader is familiar with 
the idea of angles as described in plane geometry. We have 
two lines AB and AC each ex¬ 
tending indefinitely in one direc¬ 
tion from a point A. The figure 
BAC is called the angle A or the 
angle BAC. 

A line, such as AB or AC, ex¬ 
tending in only one direction from a point is often called a 
ray. The angle BAC then consists of the two rays AB and 
AC, which are sometimes called the sides of the angle. 

2. Angles generated by a rotating ray. It is useful to 
consider an angle BAC as being generated by rotating a ray 
from the side AB to the side AC; the former is called the 
initial side, the latter the terminal side. 

Thus a hand of a clock or a spoke of a rotating wheel 
generates an angle in any given length of time. 

l 



Fig. 1 



2 


TRIGONOMETRY 


3. The general angle in trigonometry. In trigonometry 
we shall consider angles generated by rotating rays. We 
note that a ray may make one or more complete revolutions 
about the point A . An angle BA C 
may, for example, be generated by 
a rotation through a part of one 
revolution, as indicated by the 
arrow in Figure 2, or by a revolu¬ 
tion and a part of another as indi¬ 
cated by the arrow in Figure 3. 
In fact there may be any number of whole revolutions 
added to the part of a revolution. In Figure 4 an angle of 
more than three complete revolutions is shown. 



Initial Side 
Fig. 2 



In drawing figures the distinction between these angles 
is most easily made by use of curved arrows as shown, the 
arrowhead being located at the terminal side. 

We shall also distinguish between directions of rotation 
of the ray. This is most easily done by use of positive and 
negative signs, just as directions on a line are indicated in 
algebra. 







THE SIX TRIGONOMETRIC FUNCTIONS 


3 


We shall agree to call an angle 'positive which is generated 
by counterclockwise rotation; that is, rotation in the direction 
opposite to that in which the hands of a clock move. An 
angle generated by a clockwise rotation will be called negative. 

The angles in Figures 3 and 4 are positive, but those in 
Figures 5 and 6 are negative. 

4. Measurement of angles. The reader is familiar with 
the measurement of angles in terms of degrees , minutes, and 
seconds. The general angle adds no difficulty. Thus in 
Figure 7 the measure of the first angle is 90°, of the second 
is 585°, and of the third is —225°. In trigonometry we use 
angles of 0° and of any positive or negative number of degrees. 
C 



We recall that a degree is divided into 60 equal parts called 
minutes, and a minute into sixty equal parts called seconds. 

1 right angle = 90°, 

1° = 60', 1' = 60". 

Other units of measure for angles are in general use. 
Thus in some European countries, a right angle is divided 
into 100 equal parts called grades, these into 100 equal parts 
called minutes, and these in turn into 100 equal parts called 
seconds. In a later chapter we shall discuss still other 
methods of measuring angles. 

5. The protractor. A given angle may be measured 
roughly by use of a protractor. This instrument is also useful 
in drawing an angle of given magnitude. 






4 


TRIGONOMETRY 


In Figure 8 a protractor is shown in position to measure 
a given angle AOB, which is seen to be an angle of 27°. 
This figure also makes it clear how to draw a line OB making 
an angle of 27° with OA, or to draw OC making an angle of 
— 153° with OA. 



EXERCISES 

1 . Draw a triangle and measure the three angles. Find 
their sum. 

2. Draw angles of 90°, 270°, 450°, 540°, -270°, -180°, 
405°, —1080°, 855°, -675°. 

3. Draw angles whose magnitudes measured in right 
angles are 2, 4, 7, —5, 0, 3j, 7|, — 6j, — 2J, —13. 

4. With a protractor construct the following angles: 

5°; 72°; -88°; 130°; 170°; -212°; 260°; -325°; 487°; -120°. 

5. With a protractor construct the following angles: 

60°; 100°; -30°; 210°; C80°; -385°; -170°; 350°; -5°; -80°. 














THE SIX TRIGONOMETRIC FUNCTIONS 5 

6. Estimate the measure in degrees of the following angles, 
then measure with a protractor: 




7. With a protractor measure the following angles in 
degrees: 




8. An auto goes ahead far enough so that a wheel makes 
ten revolutions. As viewed from the left side of the car, 
through what angle does a spoke of a wheel turn? As 
viewed from the right? 

9. Through what angle does the hour hand of a watch 
turn in 10 hours? The minute hand? The second hand? 

10. The earth goes around the sun in a year. Through 
what angle does the line from the sun to the earth turn 
in seven months? In 2\ years? Consider the angles 
positive. 






6 


TRIGONOMETRY 


6. Directions measured from a line of reference. There 
are several methods in common use for describing a direction. 
All depend on determining the angle that a line having the 
given direction makes with some fixed line, which we call a 
line of reference. Let us explain a few methods which we 
shall use in this book. 

The one which we shall employ most takes as line of 
reference a horizontal line, or one running from left to right, 
and uses the general angle of trig¬ 
onometry to describe the angle. 
-A Thus the direction from 0 to P in 
Figure 9 is said to make an angle 
with OA of — 67§°, or 292|°, or 
any angle differing from these by 
a multiple of 360°. The direction 
from 0 to Q is 157f°, or 517J°, or 
—202§°, as measured from the line of reference OA. 

In surveying , the common practice is to use the North- 
South line as the line of reference, and state in degree meas¬ 
ure the acute angle which a ray in the given direction makes 
with this line. Thus the direction of P from 0 in Figure 9, 
called the bearing of P from 0, is South 22i° East, which is 
written S 22|° E. The bearing of Q from 0 is N 67J° W. 

In the U. S. Navy angles are measured from the North 
around through the East in the clockwise direction, in de¬ 
grees up to 360°. Thus the direction, or bearing, of P from 
0 is 157|°, and of Q from 0 is 292|°. 

7. Location of a point by distance and direction. A point 
in a plane can be located by giving its distance and direction 
from some given point. For example, surveyors can locate 
an object by saying that it is 100 ft. N 20° E from a certain 
stake. A sailor can locate a rock by stating that it has a 
bearing of 50° from a certain lighthouse, and is 1 mile from it. 

This is a very simple idea which is obviously of great prac¬ 
tical importance and is used extensively in mathematics and 




THE SIX TRIGONOMETRIC FUNCTIONS 


7 


its applications. In the next section we explain the exact 
form in which it will be employed. 

8. Polar coordinates. We choose a point 0, called the 
pole, and a line of reference, OA, called the polar axis. 
Then a point P is located by two numbers, r and d* the 
first giving the length, the second the direction of OP. 
These two numbers are called polar coordinates of P. Dis¬ 
tances and angles are measured in terms of appropriate 
units. 

If the unit of distance is the inch, then the polar coordi¬ 
nates of P in Figure 10 are (1,30°). It is customary to 
write them in parentheses, the dis¬ 
tance first and the angle second. The 
unit of angular measurement is often 
indicated but the unit of distance is 
generally not specified. 

It is to be noted that a point may 
be located by different angles in polar coordinates. The 
point P in Figure 10 has, for example, polar coordinates 
(1,390°), (1,750°), (1, -330°). 



Polar Axis 
Fig. 10 


It is sometimes convenient to use the idea of negative 
directions in measuring distances. We locate the same 
point P by going in the direction 210° a distance — 1. When 
this plan is followed, the point P has also the polar coordi¬ 
nates (-1,210°), (-1, 570°), (-1, -150°). 


EXERCISES 

A direction may be described in the three ways stated in § 6, 
which we may call ( a ) the Surveyor method, (b) the Navy 
method, and (c) the Polar Coordinate method. In the following 
tables each direction is described by some one method. For 
polar coordinates we assume that the East direction is taken as 
the polar axis. Fill in the description by the other methods. 

* q i s the Greek letter “theta.” In this book we shall also use the Greek 
letters « (alpha), /3 (beta) and 7 (gamma). 



8 


TRIGONOMETRY 



Surveyor 

Navy 

Polar Coordinate 

(a) 

N 45° E 



(6) 


135° 


(c) 



-135° 



Surveyor 

Navy 

Polar Coordinate 

(a) 

S 22i° E 



(6) 


O 

OO 


(c) 



—348f° 



Surveyor 

Navy 

Polar Coordinate 

(a) 

S 11|°E 



(6) 


to 

o 

to 

tO|M 

o 


(c) 



-202|° 



Surveyor 

Navy 

Polar Coordinate 

(a) 

N 33f°W 



(6) 


o 

pjN 

00 

u- 


(c) 



922|° 


On a sheet of -paper draw a polar axis OA , choose a convenient 
unit of length, and locate the following points: 

5. A (4,45°); 5(3,180°); <7(2,300°); D(5, - 90°); 

E( 2, -120°). 

6. A(2, 60°); 5(3,135°); C(4,270°); 5(1,405°); 

E( 2, -45°). 

7. A(—1,70°); B(—2,135°); C(-l, 180°); 5(-2, -360°); 
£(-3,-750°). 

8. A(- 2,0°); B(- 3, -90°); C(-4, -30°); 5(-1,900°); 
£( —1, -585°). 



































THE SIX TRIGONOMETRIC FUNCTIONS 


9 


D 

Fig. 11 


9. Directed lines. In the last paragraph we recalled the 
use of positive and negative directions on a line. We shall 
need to go a little further with that idea now. 

If on a given line we decide to call segments of lines meas¬ 
ured in one direction positive, and in the opposite direction 
negative, we call the line a directed line. Let AB be a 
directed line. Then if we consider the segment CD (Fig. 11) 
measured from C to ^ n 

D positive, and DC - 1 - 

measured from D to A 
C negative, we shall 
refer to the direction AB as the positive direction of the line. 
It is understood that all segments measured in the same di¬ 
rection on AB have the same sign. Thus PQ, PC, CQ, and 
DQ are all positive in the figure, while QP, CP, QC, and QD 
are negative. It is convenient to indicate the positive di¬ 
rection on a directed line by an arrowhead. 

When a unit of measure has been chosen, then the seg¬ 
ments on a directed line are measured by positive and nega¬ 
tive numbers. Thus in Figure 11, if CD is the unit of 

length the measure of CQ is 2, 
of CP is -1, of QP is -3. 

10. Projection. If we drop a 
perpendicular from a given 
point P to a given line AB, the 
foot of the perpendicular M is 
called the projection of P on 
AB. If we have a segment PQ 
of a directed line, and project P 
and Q on a directed line AB so that M and N are the pro¬ 
jections of the respective points, then the projection of PQ 
on AB is MN. This is briefly written 

Proj. on AB of PQ = MN. 

The segment MN is a directed quantity and may be either 







10 


TRIGONOMETRY 


positive or negative, or if PQ is perpendicular to AB the 
projection is zero. 

11. Location of a point by rectangular coordinates. We 

discussed in §§ 7, 8 (pp. 6, 7) one method of locating a point 
in a plane. We now give a second method with which the 
student has probably already become familiar in drawing 
graphs in earlier mathematics. 

Draw two directed lines of reference X'X and Y'Y mu¬ 
tually perpendicular, intersecting at a point 0, as shown in 
Figure 13. The lines are called the z-axis and the ?/-axis, 
the point 0 the origin. 

Having chosen a convenient unit of length, we now 
locate any point P of the plane as follows. Project P on the 



two axes, calling the respective projections M and N. If 
x is the measure of the segment OM and y of ON, then the 
numbers x and y locate P and are its rectangular coordinates. 
We call x the abscissa, and y the ordinate of P. If we regard 
MP as a segment whose positive direction is upward, we may 
write 


x = OM, y = MP, 

and call OM the abscissa, and MP the ordinate of the point P. 
It is to be noted that x and y may have either sign and that 
either or both may be zero. 






THE SIX TRIGONOMETRIC FUNCTIONS 


11 


Thus in Figure 14, 

the coordinates of P are x = 2, y — 1, 
the coordinates of P' are x = —l,y — 2, 
the coordinates of P" are # = —3 , y = — 1, 
the coordinates of P'" are # = 1, y = — 2, 
the coordinates of A are x = 3, y = 0, 
the coordinates of 0 are x — 0, y — 0. 


It is customary to write the 
coordinates of a point in paren¬ 
theses, giving the x value first. 

Thus we would write the pre¬ 
ceding more briefly: P(2, 1); 

P'(-l, 2); P"(- 3, -1); 

P"'(l,-2); A(3, 0); 0(0,0). 

An example of a use of rectan¬ 
gular coordinates would lie in a 
surveyor’s description of the loca¬ 
tion of a point as 40 yd. E and 20 yd. N of a given point. In 
effect this given point is the origin, the rc-axis is the West-East line, 
the ?/-axis the South-North line, and the coordinates are x =40, y =20. 



12. Quadrants. The axes of coordinates divide the 
plane into four parts, called quadrants. They are ordinarily 
numbered as shown in Figure 15. 
Thus the point P(2, 1) of Figure 14 
lies in the first quadrant, the point 
P'(— 1,2) in the second and so on. 

The quadrants are distinguished 
by the signs Qf the coordinates of 
points lying in them. For example, 
in the second quadrant, the abscissa 
is negative and the ordinate is posi¬ 
tive, but in the third quadrant both 
are negative. In Figure 15 the signs are shown in parentheses 
for each quadrant, the sign of x preceding that of y. 


Y , 


II 

I 

(-, + ) 

(+,+) 

0 

<+-> * 

hi 

IV 


Fig. 15 






12 


TRIGONOMETRY 


We shall have frequent occasion to draw angles with OX 
as the initial line and the origin 0 as vertex. Then if the 
terminal line falls in the first quadrant, as it does for the 
angle a in Figure 16, we shall say that the angle terminates 
in the first quadrant. If the terminal line falls in the second 
quadrant, as it does for a', we say that the angle terminates 


Y 



in the second quadrant. The angle a" terminates in the 
third quadrant, a " in the fourth. Angles which are mul¬ 
tiples of 90° do not terminate in any quadrant; they are 
sometimes called quadrantal angles. 

It should be emphasized that whenever we speak of an 
angle as terminating in a quadrant, it is assumed that the 
vertex is at 0 and OX is the initial line. 

EXERCISES 

Choose a rectangular coordinate system and locate the fol¬ 
lowing points , designating each point both by letter and by 
coordinates: 

1. A(3,l); B( 1,3); C(~ 1,3); D(-3, 1); E(- 3,-1); 
^(-1,-3); G(l, -3); H( 3,-1); 7(0,5); /( — 5,0). 

2. -4(4,3); 7?(3,-4); C(3,4); 7)(-3,4); E{- 4,-3); 
F(-3, -4); G(—4,3); 77(4,-3); 7(5,0); J(0, -5). 

3. In which quadrant does each point of Exercise 1 lie? 

4. In which quadrant does each point of Exercise 2 lie? 




THE SIX TRIGONOMETRIC FUNCTIONS 


13 


5. How could a surveyor describe by coordinates the 
location of the following points whose distances are all 
measured from a given point 0? A is 40 yd. E and 50 yd. N 
from 0; B is 50 yd. S and 40 yd. W from O; C is 70 yd. N 
from O; D is 300 yd. W and 200 yd. N from O; E is 50 rd. 
N and 10 rd. E from 0. 

6. Proceed as in Exercise 5 for the points described as 
follows: 

A is 20 rd. E and 40 rd. N from 0; B is 50 rd. S from 0; 
C is 30 rd. S and 50 rd. W from 0; D is 20 rd. N and 30 rd. 
W from 0; E is 1 mi. W and 2 mi. N from 0. 

With the ray OX of a rectangular coordinate system as 
initial line , draw the following angles, and state the quadrant 
in which each terminates: 

7. 60°; 240°; -185°; 810°; -1100°. 

8. -30°; 150°; 660°; -630°; 1000°. 


13. Changing from rectangular to polar coordinates. It 

is not difficult to see that it will be desirable to have a simple 
method of solving the following problem: Given the rec¬ 
tangular coordinates of a point, what are its polar coordi¬ 
nates? A surveyor must solve such a problem when he 
knows that a point B is 500 ft. 

East and 300 ft. North from 
a point A, and wishes to find 
the direction and length of AB. 

Let us see how such a problem 
may be solved. 

We assume that the pole O 
of polar coordinates is the ori¬ 
gin of rectangular coordinates 
and that the polar axis and the 

positive z-axis coincide. As shown in Figure 17, (r,0) are 
polar coordinates and (x,y) are rectangular coordinates of 
the point P. 





14 


TRIGONOMETRY 


The problem is, given the values of x and y, to find the 
values of r and 6. 

To find r is simple, for it is the length of the hypotenuse of 
a right triangle whose other sides are known. Hence, by 
the theorem of Pythagoras, 

r 2 = x 2 + y 2 , 

and therefore _ 

r = d= Vx 2 + y 2 . 

The positive sign is to be used when r is positive as shown 
in Figure 17; the negative sign when r is negative as shown 
in Figure 18. 

To find d is a little more difficult. We notice that 6 can 
have the same value for all points, P, P', P", etc., on the 



line OP, but must have a different value for any point S 
not on the line, as illustrated in Figure 19. Also, for these 
points P, P' } P" on the line OP, the ratio of the ordinate to 
the abscissa is always the same; that is, 

MP = M'P' = M"P" * 

OM ~ OM' ~ OM" * 

But for any point S not on the line OP the corresponding 

ratio is different; that is, 

MP . , , , RS 

OM is not equal to ^ • 

* Note that both M"P " and OM" are negative, and their ratio is positive. 










THE SIX TRIGONOMETRIC FUNCTIONS 


15 


We see then that the angle 6 is determined by the ratio 
of the ordinate of any point on its terminal side to the ab¬ 
scissa of that point, or briefly, by y/x. If some one were to 
construct a table showing what angle corresponds to each 
value of the ratio y/x, it would be possible to find 6 when y 
and x are given; for we could calculate the ratio and look 
in the table for the corresponding angle. Mathematicians 
have constructed tables for this purpose. We shall not 
make a thorough study of the methods by which such 
tables are made, but shall in the following chapters see how 
they are used. 

The ratio y/x depends on 6 for its value, and is, therefore, 
in mathematical language, a function of 6. It is called the 
tangent of 0, which is written in abbreviated form tan 6 . 
Thus, by definition, 

, „ y ordinate 

tan 0 = - = —r—:- 

x abscissa 

The origin of the name “tangent of 0” may be seen as follows. 
Draw a circle with center at 0 and unit radius. Let the point of 
intersection of the circle and the posi¬ 
tive x-axis be A (Fig. 20). Draw a 
line AC tangent to the circle at A. 

Let T be the point of intersection of the 
terminal side of 0 with the tangent 
line AC. Then if P(x,y ) is any point 
of the terminal side, 

y MP AT Arr 
tan e - X~ OM - OA ~~ AT ’ 

since 0A = 1. Hence the tangent of 0 
is the measure of the length of the segment cut off on the tangent line 
A C by the terminal side of 0. The directed segment A T is called the 
line value of tan 0. We shall discuss line values in Chapter IV, § 54. 

14. Changing from polar to rectangular coordinates. 

We have just indicated how, by introducing the^trigono- 








16 


TRIGONOMETRY 


metric function tan 0, we may calculate the polar coordi¬ 
nates (r,0) of a point P when its rectangular coordinates 
(x,y) are given. We shall now see how the problem of cal¬ 
culating x and y when r and 0 ard given leads to the intro¬ 
duction of two new trigonometric functions. 

We start by observing that for all points on the line OP, 
such as P, P', P" (Fig. 19) the ratio of ordinate to distance 
is the same; that is, 

MP M'P' _ M"P" * 

~OP ~ OP' ~ OP" 

For a point S not on the line the corresponding ratio RS/OS 
has in general a different value. The ratio of ordinate to 
distance is thus a function of 0, and is called sine of 0, which is 
written sin 0. Thus 

. . n y ordinate 

^ sm r distance 


Mathematicians have made tables from which we can 
obtain the value of sin 0 when 0 is given. When r is also 
given we can therefore find y from the equation, 

y — r sin 0, 


which is equivalent to (1). 

The ratio of abscissa to distance is also a function of 0. 
It is called cosine of 0, and is written cos 0. Thus 


( 2 ) 


cos 0 


x _ abscissa 
r distance 


Tables for this function are available, so that when 0 is 
given cos 0 may be found. Hence when r and 0 are known, 
we find x by the formula 

x — r cos 0, 


which is equivalent to (2). 


* Note that M"P" and OP" are both negative. 






THE SIX TRIGONOMETRIC FUNCTIONS 


17 


15. The six trigonometric functions. We have defined 
three functions of an angle 6. To restate the definitions, 
let .a line through the origin make an angle 6 with the positive 
x-axis and let P be any point on the line. If the rectangular 
coordinates of P are ( x,y ), and the polar coordinates are 
(r,0), then 



V 

sine of 0 = sin 0 = - 
r 

x 

cosine of 0 = cos 0 = 

r 

y 

tangent of 0 = tan 0 = ^ 


ordinate 
distance ’ 
abscissa 
distance ’ 
ordinate 
abscissa 


These three functions and their reciprocals are known as the 
six trigonometric functions. The reciprocals are: 


cotangent of 0 = cot 0 = 


secant of 0 = sec 0 = - 
x 


cosecant of 0 = esc 0 = 


abscissa 
ordinate , 
distance 
abscissa ’ 
distance 
ordinate 


In using these definitions it is generally most convenient 
to choose P so that r is positive. We shall hereafter assume 
that this is done except where noted. 

The preceding definitions should be thoroughly memo- 


















18 


TRIGONOMETRY 


rized, for trigonometry is essentially a study of these func¬ 
tions and their applications. 

Three other functions are sometimes encountered in applica¬ 
tions of trigonometry. These are called the versed sine , coversed 
sine, and haversine of 0, written vers 0, covers 0, and havers 0, respec¬ 
tively. They are defined by the relations 

vers 0 = 1 — cos 0, 

covers 0 = 1 - sin 0, 

_ 1 - cos 0 

havers 0 = —-- • 

L 

It is at once apparent that for two angles which differ 
by 360°, 720° or any other positive or negative multiple of 
360°, the values of the trigonometric functions will be exactly 
the same, since the same point P may be used for both 
angles (see Fig. 22). 

16. Algebraic signs of the functions. It is to be noted 
that x and y may be either positive or negative, depending 




upon the angle 6. It follows that a function is positive for 
some angles and negative for others. 

Consider, for example, an angle 6 terminating in the 
second quadrant (Fig. 23). We have agreed to choose P 
so that r is positive; then x is negative and y positive. 








THE SIX TRIGONOMETRIC FUNCTIONS 


19 


Hence 


sin d = - 
r 

cos 6 = — 
r 

tan 0 — - 
x 



+ 

+ 


That is, sin 6 is the ratio of two 
positive numbers and is therefore 
positive; cos d is the ratio of a neg¬ 
ative to a positive and is therefore 
negative; and similarly for tan 6. 

The signs of the functions depend 
upon the quadrant in which the 
angle terminates. A discussion like 
the preceding gives results indicated 
in Figure 24. 


Y 

Sin) , 
Cscf + 

Others — 

All F 

0 

X 

Tan ) , 

Cos ) 

Cot ; + 

Sec [ + 

Others — 

Others — 


Fig. 24 


EXERCISES 


The rectangular coordinates of the following points are given; 
find for each the value of r, sin d, cos d, and tan 6. Draw a 
figure for each point. 

1. A (3, 4); £(-3,4); C(-4,-3); D(4,-3); £(-1,0). 

2. A (5,12); £(-12,5); C(-12,-5); £>(3,-4); £(3,0). 


For each of several points the distance r is 10. Find the 
rectangular coordinates of each when sin 6 and cos 6 are given 
as follows. Draw a figure for each point. 


Point 

A 

B 

c 

D 

E 

sin 6 

3/5 

-4/5 

-5/13 

0 

-1 

cos 6 

4/5 

3/5 

-12/13 

1 

0 

Point 

A 

B 

c 

D 

E 

sin 0 

5/13 

-3/5 

5/13 

0 

1 

cos d 

12/13 

-4/5 

-12/13 

-1 

0 










20 


TRIGONOMETRY 


5. Find the values of the six trigonometric functions of 9 
when the point A (6, 8) is on the terminal line. Likewise 
for each of the following points: B( — 7, 24); C( —8, —15); 
D( 21, -20); E{ 33, 56). 

6. Proceed as in Exercise 5 for the points: ^4.(16, 12); 
£(—8,15); C(-21,-20); £>(28, -45); #(11,60). 

7. Write out the discussion of signs of the functions for 
angles terminating in the third quadrant. 

8. Write out the discussion of signs of the functions for 
angles terminating in the fourth quadrant. 



Fig. 25 



















































































































THE SIX TRIGONOMETRIC FUNCTIONS 


21 


17. Values of the functions by measurement. Approxi¬ 
mate values of the trigonometric functions of an angle can 
be found by construction of the angle, measurement of co¬ 
ordinates of a point on the terminal line, and calculation of 
the ratios. By taking a succession of angles and proceeding 
in this way, we can make a table of values of the functions. 
While this method is not used by mathematicians in making 
a table, it gives an instructive exercise. 

In Figure 25 we have a circle of radius 50 mm. graduated 
to 5° intervals. To find, for example, the functions of 20° 
we take P as the point of intersection of the 20° line and 
the circle. The vertical and the horizontal fines are 2 mm. 
apart; then from the figure we read x = 47, y = 17, r = 50. 
Hence 


sin 20° = E = m, 
o U 

so 

CSC 20° = jy = 2.9, 

cos 20° = = .94, 

50 

sec 20° =^= 1.1, 

tan 20° = = .36, 

cot 20° = ^ = 2.8. 


The results are given to two significant figures, which is 
all that should be used since measurements of x } y, and r 
are no more accurate. (See § 34, p. 46.) 

By repeated use of this method we make the table given 
on the next page. 

To illustrate how to read the table let us find cos 55°. 
We go down the column headed “Angle” to 55°, then across 
the row to the column headed “Cos”; we find .57 which is 
the value of cos 55°. 

No value is given for cot 0° since 50/0 has no value 
(no number multiplied by 0 gives 50). A similar remark 
applies to esc 0°, tan 90°, and sec 90°. 


22 


TRIGONOMETRY 


Angle 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0° 

.00 

1.00 

.00 


1.00 

li .5 

5° 

.09 

1.00 

.09 

l i. i 

1.00 

10° 

.17 

.98 

.18 

5.67 

1.02 

5.76 

15° 

.26 

.97 

.27 

3.73 

1.04 

3.86 

20° 

.34 

.94 

.36 

2.75 

1.06 

2.92 

25° 

.42 

.91 

.47 

2.14 

1.10 

2.37 

30° 

.50 

.87 

.58 

1.73 

1.15 

2.00 

35° 

.57 

.82 

.70 

1.43 

1.22 

1.74 

40° 

.64 

.77 

.84 

1.19 

1.31 

1.56 

45° 

.71 

.71 

1.00 

1.00 

1.41 

1.41 

50° 

.77 

.64 

1.19 

.84 

1.56 

1.31 

55° 

.82 

.57 

1.43 

.70 

1.74 

1.22 

60° 

.87 

.50 

1.73 

.58 

2.00 

1.15 

65° 

.91 

.42 

2.14 

.47 

2.37 

1.10 

70° 

.94 

.34 

2.75 

.36 

2.92 

1.06 

75° 

.97 

.26 

3.73 

.27 

3.86 

1.04 

80° 

.98 

.17 

5.67 

.18 

5.76 

1.02 

85° 

1.00 

.09 

11.4 

.09 

11.5 

1.00 

90° 

95° 

1.00 

1.00 

.00 

-.09 

-11.4 

.00 

-.09 

-11.5 

1.00 

1.00 

100° 

.98 

-.17 

-5.67 

-.18 

-5.76 

1.02 


18. Applications. By use of the preceding table extended 
up to 360° we can get approximate solutions of certain 
problems. Later we shall see how more accurate results 
can be obtained. 

Examples . — 1. The rectangular coordinates of a point 
are (11, 60). Find the polar coordinates. 

We have 

x = H ; y = 60, r = V x 2 + y 2 = 61, 


tan 0 = - = 5.45. 
x 

In the table, in the “Tan” column we do not find 5.45 but a near 
value is 5.67, which occurs in the row with the angle 80°. Hence 
d = 80° approximately. 

The polar coordinates as thus determined are (61, 80°). 

























THE SIX TRIGONOMETRIC FUNCTIONS 


23 


2. Polar coordinates of a point are (70, 100°). Find the 
rectangular coordinates. 

Since 

y % 

sin 0 = -, cos 0 = -, 
r* r 

we have 

y = r sin e, x = r cos 0. 

From the table, sin 100° = .98, cos 100° = —.17. Hence 
y = 70 X .98 = 68.6, x = 70 X (-.17) = -11.9. 

To two significant figures the rectangular coordinates are ( — 12, 69). 


EXERCISES 

Verify by measurement and calculation the values given in 
the table in § 17 for the following angles, using Figure 25: 

1. 10°, 40°, and 70°. 2. 30°, 50°, and 80°. 

3. 0°, 60°, and 100°. 4. 45°, 90°, and 95°. 

5. Extend the Sin and Cos columns of the table in § 17, 
using angles 120°, 135°, 150°, 165°, 180°, 195°, 210°, 225°, 
240°, 255°, 270°, 285°, 300°, 315°, 330°, 345°, 360°, 375°. 

6. Extend the Sin and Cos columns of the table in § 17, 

using angles which are multiples of 10° up to 400°. (Much 
work may be saved by comparing values of functions of angles 
terminating in other quadrants with those of angles termi¬ 
nating in the first quadrant.) v 

Find the polar coordinates (< approximate) from the given 
rectangular coordinates , for the points: 

7. A(3, 4); £(-5,12); C(-16, 12). 

8. A (4, 3); £(-8,15); C(-ll,60). 

Find the rectangular coordinates ( approximate) from the 
given polar coordinates for the points: 

9. A(10, 20°); £(55,80°); C(35, 100°). 

10. A(20, 15°); £(30,75°); C(40, 95°). 


24 


TRIGONOMETRY 


11. A boat is sailing a course (p. 6) of 350°. When it is 
at a point A , a rock R bears due West, when at B due South. 
The distance from A to B is 3 mi. How far is the boat from 
the rock when at A and when at B ? 

12. A surveyor measured a line diagonally across a rec¬ 
tangular field; the bearing of the line (p. 6) was N 30° E; 
its length was 300 yd. The sides of the field ran due East 
and due North respectively. What was the perimeter of 
the field? 

13. A surveyor runs a line due East 200 yd. from A to B 
then due North 300 yd. from B to C. How far is it from A 
to C, and what is the direction? 

14. A navigator wishes to sail from A to B. From the 
differences in longitudes of A and B he knows that B is 
40 mi. West of A; and from the difference in latitudes he 
knows that B is 25 mi. North of A. What is the distance 
and direction from A to B, assuming the surface of the 
water to lie in a plane? 


19. Functions of 45° 



, 135°, 225°, and 315°. There are 
some angles for which the exact 
values of the functions can be found 
by direct use of propositions of 
geometry. They are of enough im¬ 
portance to receive special atten¬ 
tion. 

For a 45° angle choose a point P 
such that x = 1. It is then readily 
shown that y — 1, and by the 
theorem of Pythagoras r — \/2. 
Hence 


sin 45° = = 

V2 

tan 45° = 1, 
sec 45° = V2, 


V2 
2 ’ 


cos 45° = —= 

V2 

cot 45° = 1, _ 
esc 45° = V2. 


V2 

2 ’ 





THE SIX TRIGONOMETRIC FUNCTIONS 


25 


For an angle of 135°, take P such that x — —1; then we 
see (Fig. 27) that y — 1, r — \/2, and hence 


sin 135° — 

sml65 2 ’ 

tan 135° = -1, 
sec 135° = -V2, 


cos 135° = ■ 

V2 

cot 135° = -1, 
esc 135° = V2. 


V2 
2 ? 



Similarly for 225° we find 
sin 225° = ~ , cos 225° = ~ = 

V2 2 V2 2 ’ 

tan 225° = 1, cot225°=l, 

sec 225° = — s/2, esc 225° = -\/2. 

20. Functions of 30°, 60°, and 120°. Let ABC be an 

equilateral triangle, whose sides are each of length 2 (Fig. 28). 
Drop a perpendicular from C to AB; let 
D be the foot; then D bisects AB. The 
angles of the triangle ADC are 30°, 60°, 

90° (Why?). The sides opposite those 
angles are respectively 1, V3, 2 (Why?). 

A triangle with angles 30°, 60°, 90° occurs 
in each of the following figures. To find Fig. 28 












26 


TRIGONOMETRY 


the values of the functions of 30°, we draw Figure 29. From 
the definitions of the functions, we have 



tan 30° = , cot 30° 

v3 6 

sec 30° = -?= = ^^, esc 30° 
v3 3 

From Figure 30, we have 


V3 
2 ’ 

V3, 

2 . 




sin 60° = > 

tan 60° = V3, 
sec 60° = 2, 


cos 60° = ~ , 
cot 60° = ’ 

„„„ 2 V3 

esc 60 = —g— 













THE SIX TRIGONOMETRIC FUNCTIONS 


27 


For the angle 120°, we have from Figure 31, 
sin 120° = 

tan 120° = -V3, 
sec 120° = -2, 

The student may draw figures similarly for 150°, 210°, 
240°, 300°, and 330°, and thus derive exact values of the 
functions of these angles. 

21. Functions of 0°, 180°, and 90°. For an angle of 0° 
the terminal line coincides with the initial line; there has 
been no rotation. A point P on 


the terminal line lies on the z-axis 
and we have 

Y 


II 

II 

p 

1 


ii ii 

Hence 

sin 0° = - = 0, 
r 

O 

M X 


7* Fig. 32 

cos 0° = - = 1, 
r 


0 T 

tan 0° = - = 0, cot 0° = -, impossible, 

sec0° = - = 1, esc 0° = -, impossible. 

Two of the definitions, those for cot 0° and esc 0°, lead to 
division by zero; but since no number times zero gives r, 
there is no value for these functions of 0°. 

In § 55 we shall discuss values of functions of angles near 
0°; for such angles the cotangent and cosecant are very 
large. 


cos 120° = -i 

z 

cot 120° = 

o 

CSC 120° =^r- 





28 


TRIGONOMETRY 


For 180° we have (Fig. 33) 
x = —r, 


Hence 

sin 180° = - = 0, 
r 

tan 180° = - = 0, 

V 


sec 180° = — = -1, 
— r 


y = 0 . 

cos 180° = —= -1, 
r 

— T 

cot 180° = — , impossible, 

T 

esc 180° = -, impossible. 



Y 

✓-V 

a 

II II 

S O 

p(§=o r ) 



M 

OX 0 

M X 

Fig. 33 

Fig. 

34 


For 90° we have (Fig. 34) 
x = 0, 

Hence 


y = r. 


sin 90° = - = 1, 
r 

cos 90° 

r 

tan 90° = -, impossible, 

cot 90° 

r 

sec 90° = ^, impossible, 

esc 90° 


0 

r 

0 

r 

r 

r 


0 , 

0, 

1. 


EXERCISES 

From suitable figures find the exact values of the functions 
of the following angles: 

1. 315°; 150°; 240°; 270°; 360°. 

2. -45°; 210°; 300°; -90°; 540°. 

3. -135°; -60°; 690°; -225°; 720°. 

4. -330°; 780°; -405°; -810°; 1080°. 

5. Show that 1/2 is the exact value of 

sin 60° cos 330° + cos 60° sin 330°. 






THE SIX TRIGONOMETRIC FUNCTIONS 


29 


6. Show that V3/2 is the exact value of 

cos 150° cos 240° — sin 150° sin 240°. 

7. Find the exact value of 

sin 30° cos 150° - cos 30° sin 150°. 

8. Find the exact value of 

cos 45° cos 210° + sin 45° sin 210°. 


22. Problems in which a function is given. We give three 
illustrations of types of problems in which the value of a 
function of an angle is given. 

Examples. — 1. Given sin0 = 3/5; construct possible 
angles 6 and find values of the other functions of d. 

We have y/r =3/5. We locate a point P for which y = 3, 
r = 5 (y = 6, r = 10 would serve as well). To do this draw first 
a circle with center at 0 and radius 
5; at every point of this circle, 
r = 5. Draw next a line parallel to 
the a:-axis, 3 units above it; at every 
point of this line, y = 3. At the 
points of intersection of the line and 
. the circle we have y = 3, r = 5. 

Call these points Pi and P 2 . Draw 
OP i and OP 2 ) either of these lines 
may serve as terminal line of the 
angle 0. There are two such angles 
which are positive and less than 360°. 

Call them di and 0 2 (Fig. 35). Since 

x 2 + y 2 = r 2 , 

we have x = ±4; for Pi, x = 4; for P 2 , x = —4. Then 
sin 0i = 3/5, cos 0 X = 4/5, 

tan 0i = 3/4, cot 0i = 4/3, 

sec 0i = 5/4, esc 0i = 5/3, 



t 


sin 0 2 = 3/5, 
tan 0 2 = —3/4, 
sec 0 2 = —5/4, 


cos 0 2 = —4/5, 
cot 0 2 = —4/3, 
esc 0 2 = 5/3. 


and 







30 


TRIGONOMETRY 


Are there solutions other than 0i and 0 2 ? The answer is in the 
negative, if we restrict ourselves to positive angles less than 360°. 
For, taking r positive, we must have y positive, and 0 must terminate 
in the first or the second quadrant; and it is easy to see that for 
any point P not on OPi or OP 2 the ratio y/r cannot be 3/5. 

2. Given tan d = 5/12; construct possible angles 6, and 
find values of all functions of 0. 



Since tan 0 = y/x, we locate Pi and P 2 where x = 12, y = 5, and 
where x H —12, y = —5, respectively. Then either OPi or OP 2 
may serve as terminal line for 0. Let 0i have the terminal line OP i 
and 0 2 the terminal line OP 2 . Since 

r 2 = x 2 + y 2 , 

we have r = 13. The values of the functions of 0i and 0 2 may now 
be written at once; we leave this to the reader. 

3. Express all of the 
trigonometric functions 
in terms of sin 0. 

Take r = 1, y = sin 0, and 
proceed as in Example 1. 
Since 

x 2 + y 2 = r 2 , 
we have 

, x 2 = 1 — sin 2 0, 

x = ±Vl — sin 2 $ 



Fig. 37 













THE SIX TRIGONOMETRIC FUNCTIONS 


31 


If we assume that sin 0 is positive, we see that there is an angle 0i 
terminating in the first quadrant, for which x = Vl — sin 2 0, and 
another angle 0 2 t erminating in the second quadrant for which 
x = — Vl — sin 2 0. From Figure 37, we have 


sin 0i = sin 0, 

sin 0 

tan 0i = . .: •- 

vl - sin 2 0 


sec 0i = 


1 

Vl — sin 2 0 ’ 


cos 0i 
cot 0i 

CSC 0i 


= Vl — sin 2 0, 

_ Vl — sin 2 0 
sin 0 * 

1 

~ sin 0 ’ 


For the functions of 0 2 a negative sign is placed before the radical 
in each corresponding formula for 0i. 

If sin 0 were negative the angles 0i and 0 2 would terminate in the 
fourth and third quadrants respectively, but the preceding equa¬ 
tions would still be true. 


EXERCISES 

Find the values of the other functions of 9, when it is given 
that: 


1. cos 9 = 12/13. 

3. sec 9 = —25/7. 
5. tan0 = —21/20. 
7. cos 9 = —1/3. 

9. sec 9 = — 1. 


2. cot 9 = 8/15. 

4. esc 9 = —17/8. 
6. sin 9 = —35/37. 
8. cot 9 = —4/7. 
10. esc 9 = 2. 


Express all of the trigonometric functions in terms of the 
following: 

11. cos 9. 12. tan 9. 13. cot 9. 14. sec 9. 


★23. Projection on coordinate axes. Consider a directed 
line A B which makes an angle 9 with the x-axis of a system of 
rectangular coordinates, and let CD be a segment of AB 
(Fig. 38). On a directed line through 0 making the angle 
9 with the z-axis, take OP = CD. Then the projection 











32 


TRIGONOMETRY 


C'D' of CD on the z-axis equals OM, the projection of OP 
on the a>axis. This may be written 

Proj* CD - Proj* OP = OM. 



The student should verify these formulas not only for the 
angle of Figure 38 but also for that of Figure 39 and other 
figures. 

★ 24. Vectors. Com¬ 
ponents. Resultants. 

A quantity which may 
be represented by a 
directed line segment 
CD is often called a 
vector quantity. Thus 
force, velocity, and ac¬ 
celeration are vector 
quantities. The pro¬ 
jections of a vector quantity on the x- and y -axes are called 
components of the vector. 

If F is the magnitude of a force which makes an angle 9 
with the x-axis, and if F x and F y are the components of the 
force, then, by formulas (1) and (2) of § 23, 

F y = F sin 9. 



F x = F cos 9, 










THE SIX TRIGONOMETRIC FUNCTIONS 


33 


Similar formulas hold for velocity and acceleration. 

If the components F x and F y are given, the vector F is 
c alled the r esultant. It is seen that the magnitude of F is 
VF X 2 + F y 2 . The direction of F is given by the angle 0, 
where tan 0 = F y /F x . 

If two forces, represented in magnitude and direction by 
AB and AC, act on a particle at A, they are equivalent to a 
single force, called the resultant force , acting on the particle. 




Fig. 41 


The magnitude and direction of this resultant are repre¬ 
sented by the diagonal AD of the parallelogram of which AB 
and AC are two sides. This principle is known as the 
Parallelogram Law of Forces. A similar law holds for 
velocities and accelerations. 

Example. — A force of 20 lb. acts at an angle of 40° with 
the horizontal. What two forces, one horizontal, the other 
vertical, would be equivalent? 

In the vertical plane of the force, let the 
z-axis be horizontal, the y -axis vertical. 

Then F y 

F x = 20 cos 40°, F y = 20 sin 40°, 

Using the table on page 22, we have 
cos 40° = .77, sin 40° = .64. 

TTpp pp 

F x = 15.4 lb., F y = 12.8 lb. 

These values are of course approximations. 













34 


TRIGONOMETRY 


EXERCISES 

1. Draw a figure similar to those in § 23, making 0 an 
angle terminating in the second quadrant, and verify for¬ 
mulas (1) and (2). Note that the signs as well as the 
magnitudes are correct. 

2. Proceed as in Exercise 1, making 0 an angle terminating 
in the fourth quadrant. 

3. If a boat is traveling N.E. with a speed of 20 mi. 
per hr., what is the component of its velocity in the East¬ 
ward direction? In the Northward direction? 

4. If a boat is making 30 knots per hour on a course of 
70° (see §6, p. 6), what are its components of velocity in 
the Eastward and the Northward directions respectively? 

5. A swimmer in crossing a stream puts forth efforts which 
in still water would carry him directly across at 3 mi. per hr. 
If the current is 4 mi. per hr. what is the actual direction and 
speed of the swimmer? 

6. An airplane heads West, running so that in still air 
it would have a speed of 100 mi. per hr. There is a wind 
from the South blowing wfith a speed of 40 mi. per hr. What 
is the actual direction and speed of the airplane? 

7. A force of 12 lb. acts vertically upward and another of 
20 lb. acts horizontally on a particle. What are the magni¬ 
tude and the direction of the single force equivalent to the 
two? 

8. A force of 2 tons acts horizontally, another of 5 tons 
acts vertically on a particle. What are the magnitude and 
the direction of the resultant force? 

9. A boat sails on a course of 130° (§ 6, p. 6) with a 
speed of 12 knots per hour. What are the Eastward and 
Northward components of its velocity? 

10. A surveyor runs a line 600 yd. N 10° W from A to B. 
How far East and how far North is B from A? 


CHAPTER II 


RIGHT TRIANGLES 


25. The problem of solving a triangle. The three sides 
and three angles of a triangle are called its six parts. If 
some of the six parts are given it may be possible to calculate 
the others. To do so is to solve the triangle. 

In the present chapter we shall discuss the solving of tri¬ 
angles one of whose angles is a right angle, or in other words, 
the solving of right triangles. 

26. Functions of an acute angle of a right triangle. We 
shall make use of the trigonometric functions defined in 
§ 15 (p. 17) but we shall find it con¬ 
venient here to word the definitions 
somewhat differently. 

Let ABC be a right triangle, with 
C the right angle. Let a, b, c be the 
sides opposite the angles A, B, C, re¬ 
spectively. For the angle A we 
shall call the side a the opposite side , and b the adjacent 
side. 

From the definitions of § 15, we see that 



a _ opposite side 
c hypotenuse ’ 
b _ adjacent side 
c hypotenuse ’ 
_ a _ opposite side 
an — b adjacent side ’ 
b _ adjacent side 
a ~ opposite side 5 
35 


sin A = 


cos A 


cot A 








36 


TRIGONOMETRY 


. c hypotenuse 
seC ~ b ~ adjacent side ’ 

esc a - c - h yp Qtenuse 

a opposite side’ 

These formulas should be memorized. 

27. Functions of complementary angles. For the angle 
B, the side b is the opposite side, and a the adjacent side. 
Hence 


. „ b 

n a 

sin B = - , 

cos B = - 

c ’ 

c 

tan B = -, 

cot B =~ 

a’ 

b 

D C 

sec B = -, 

D c 

•esc B = . 

a ’ 

b 


By comparing with the formulas of § 26, we see that 

sinB = cos A, cos B = sin A, 

(1) tan B — cot A, cot B = tan A, 

sec B = esc A, esc B = sec A. 

The angle B is the complement of the angle A, that is, 
B = 90° — A; it might be written co. A. The first of these 
equations could be written 

sin co. A = cos A, 

the others similarly. 

If we call the following pairs of functions cojunctions of 
each other: 

sine and cosine, 
tangent and cotangent, 
secant and cosecant, 

then the six formulas (1) are given by the rule: 

A Junction of the complement of an acute angle equals the 
cofunction of the angle. 




RIGHT TRIANGLES 


37 


For example, since 30° and 60° are complements, we have 
sin 30° = cos 60°, cos 30° = sin 60°. 

These relations are verified by reference to the values given in 
§20 (p. 25). 

28. Tables of values of functions of acute angles. To 

solve right triangles we must know the values of functions 
of acute angles. A small table of values was worked out 
in §17. The values were given only approximately — to 
two or three figures. On page 4 of the Tables, more accu¬ 
rate values are given, and values for more angles. Let us 
see how the Tables are read. 

Angles 10' apart are given from 0° up to 45° in the first 
column of pages 4-8, and from 90° down to 45° in the last 
column. The values of the functions are given in successive 
columns. For angles given at the left, we read the name of 
the function at the top of the columns; for angles at the 
right, we read the functions at the bottom of the columns. 
It will be observed that the arrangement of the Tables is 
such that the value of a function of an angle may also be 
read as the value of the cofunction of the complementary 
angle. 

Examples. — 1. To find sin 4° 40' we look on page 4, go down 
the left-hand column headed “Degrees” to 4° 40' and across to the 
column headed “Sin”; the entry is 814, which means that sin 4° 40' 
= .0814, the first digit, in this case 0, being given only at intervals 
in this table. 

2. To find cot 14° 10' we turn to page 5, go down the first 
column to 14° 10', across to the column headed “Cot” and read 
3.962. Thus cot 14° 10' = 3.962. 

3. To find cos 66° 20', turn to page 6, go up the last column to 
66° 20', across to the column with “Cos” at the bottom, and read 
.4014. That is, cos 66° 20' = .4014. 

4. Given that tan A = .7954; to find A. Look down the column 
headed “Tan” to entry .7954; go across to the first column and 
find A = 38° 30'. 


38 


TRIGONOMETRY 


5. Given that sin A = .9080; to find A. This number is not 
found in a column with “Sin” at the top, but on page 6 with Sin 
at the bottom we find values near .9080. This number lies between 
two given in the Table, namely .9075 and .9088, being nearer the 
former. Hence, going across to the last column, we find that A is 
nearly 65° 10'. 


EXERCISES 

Find values of the following: 


1. 

sin 33° 

40'; 

cos 17° 20'; 


tan 18° 

O'; 

cot 42° 50'; 


sec 12° 

10'; 

esc 8° 20'. 

2. 

sin 28° 

O'; 

cos 40° 50'; 


tan 44° 

10'; 

cot 6° 10'; 


sec 40° 

40'; 

esc 45° O'. 

3. 

sin 57° 

10'; 

cos 68° 20'; 


tan 88° 

50'; 

cot 46° 10'; 


sec 80° 

O'; 

esc 75° 10'. 

4. 

sin 88° 

10'; 

cos 46° 50'; 


tan 60° 

O'; 

cot 85° 40'; 


sec 50° 

10'; 

esc 66° 20'. 

Find the angle A in 

each of the f 

5. 

sin A 

= .5616; 

cos A = 


tan A 

= .1110; 

cot A = 

6. 

sin A 

= .1132; 

cos A = 


tan A 

= .9490; 

cot A = 

7. 

sin A 

= .7826; 

cos A = 


tan A 

= 8.345: 

; cot A = 

8. 

sin A 

= .9613 

; cos A = 


tan A 

= 4.705 

; cot A = 


.8526; 


.2278; 


RIGHT TRIANGLES 


39 


Find A to the nearest 10' in each of the following: 


9. sin A 

= .2538; 

cos A = .9953; 

tan A 

= 3.598; 

cot A = .1222. 

10. sin A 

= .9904; 

cos A = .2692; 

tan A 

= .5180; 

cot A = .9413. 


29. Interpolation. In finding the value of a function of 
an angle, such as 17° 23', which is not given in the Table 
but lies between two angles that appear, we use the method 
of interpolation, as illustrated in Examples 1 and 2 below. 
In Examples 3 and 4 the method is applied in finding the 
angle when the value of one of its functions is given. 

Examples. — 1. To find sin 17° 23'. 

The given angle, 17° 23', is three-tenths of the way from 17° 20' 
to 17° 30'. We assume that sin 17° 23' is three-tenths of the way 
from sin 17° 20' to sin 17° 30'. The sine of 17° 23' will then be 
obtained by taking 3/10 of the amount by which sin 17° 30' exceeds 
sin 17° 20', and adding this correction to sin 17° 20'. Hence 

sin 17° 23' = sin 17° 20' + 3/10 (sin 17° 30' - sin 17° 20') 

= .2979 + 3/10 (.0028) = .2979 + .00084 
= .2987 approximately. 

Since the Tables give values to only four places, we give only four 
places in our value of sin 17° 23'. This amounts to calling the 
correction .0008 instead of .00084. We would have used ,.0008 for 
any correction greater than .00075 and less than .00085. It is 
customary to disregard the decimal point in the tabulated values 
and call the tabular difference 28 instead of .0028, and the correction 
8 instead of .0008. 

Another way to explain the preceding interpolation is to state 
that we have assumed that when an angle increases, its sine increases 
proportionally; or, in other words, that differences between angles 


40 


TRIGONOMETRY 


are proportional to differences between their sines. For the ex¬ 
amples just solved the accompanying small table indicates these 
differences. We thus have 


x_ _3_ 

28 “ 10* 

Then x = 8.4 = 8 approximatel 
and 

sin 17° 23' = .2979 + .0008 
= .2987. 



Angle 

Sin 


“ ri7° 20' 

.2979“ 

10 

Ll7° 23' 



17° 30' 

.3007 


28 


The assumption just made that differences between angles are 
proportional to differences between the values of a function of those 
angles is not exactly true, but it gives rise to errors which are negli¬ 
gible when the differences involved are small. 


2. To find cot 17° 15'. 

Angle 

Cot 

From the little table at the right 

“ ri7° io' 

3.237“ 

we have 10 

L17° 15' 

-_ 

x = 5/10 X 33 = 16.5. 

17° 20' 

3.204 


The correction x could be called either 16 or 17. In all such cases 
we shall arbitrarily use the even number; here we take x = 16. 
We note that the cotangent decreases when we go from 17° 10' to 
17° 20'; hence the correction, which should take us 5/10 of the way 
from cot 17° 10' to cot 17° 20', must be subtracted from the former. 
We have 

cot 17° 15' = 3.237 - .016 = 3.221. 


3. Given tan A = .4361. To find A. 

We find that the angle A lies between 23° 30' and 23° 40', 


shown to the right. By the prin¬ 
ciple of proportional differences 
we have 

13 1 so 10 


x = 35 X 10 


Hence 


130 _ 37 

35 ~ 6 ‘ 7 ' 


A = 23° 30' + 4' 


Angle 

Tan 

o 

CO 

o 

CO 

<N 

1- 

.4348“] 

L a 

.4361J 

23° 40' 

.4383 

= 23° 34'. 



13 


35 


















RIGHT TRIANGLES 


41 


4. Given cos A = .4100. To find A. 
Proceeding as before we have 

20 


x = ~ X 10 = 8. 


Hence 


A = 65° 48'. 



Angle 

Cos 


" |~65°40' 

.4120"] 

10 

X l A 

.4100 J 


. 65° 50' 

.4094 


20 


26 


EXERCISES 

By interpolation find values of the following: 

1. sin 32° 27'; cos 22° 31'; tan 18° 47'. 

2. sin 5° 14'; cot 42° 8'; sec 22° 33'. 

3. sin 72° 15'; tan 61° 18'; esc 82° 12'. 

4. tan 81° 9'; sec 54° 54'; cot 67° 8'. 

Use interpolation to find the value of A to the nearest minute 
in the following equations: 


5 . 

6 . 

7. 

8 . 
9. 

10 . 

11 . 

12 . 


sin A 
tan A 
cos A 
sin A 
sin A 
sin A 
tan A 
tan A 


.5306; 

.6530; 

.8300; 

. 1200 ; 

.9926; 

.7671; 

1.314; 

6.923; 


cot A 
cot A 
tan A 
cos A 
cot A 
cos A 
cot A 
cos A 


3.460. 

2.380. 

.5000. 

.9601. 

.7302. 

.2581. 

.7040. 

.5610. 


30. Solution of typical right triangles. In the present 
section we shall consider triangles with sides and angles 
lettered as in § 26 (p. 35). We note that C = 90°. 

When numerical values are given for two of the parts 
A, B, a, h, c, if one at least is a side it is possible with the 
aid of Tables I and II to solve the triangle.* We use the 
formulas of § 26, together with the propositions of geometry 
expressed by the formulas 

(1) a 2 + b 2 = c 2 , 

(2) A + B = 90°. 

* In § 102 (p. 185), right triangles will be solved by means of logarithms. 






42 


TRIGONOMETRY 


Examples . — 1. Given 4 = 40° 20', c — 25. To find B } a , 5. 
From (2) we have 

B = 90° - 4 =49° 40'. 

Since sin 4 = a/c, we have 

a = c sin 4 = 25 X .6472 = 16.18. 

And since cos 4 = b/c, we have 

6 = c cos 4 = 25 X .7623 = 19.06. 


2. Given 4 = 31° 30', b = 2.5. To find B, a , c. 
From (2) we have 

B = 90° - 4 = 58° 30'. 


Since tan 4 = a/b, we have 

a = b tan 4 = 2.5 X .6128 = 1.532. 


And since sec 4 = c/b, we have 

c = 6 sec 4 = 2.5 X 1473 = 2.932. 

Instead of sec 4 we might have used cos 4 to find c. Since 
cos 4 = b/c, we have 


b 

cos 4 


2.5 

.8526 


2.932. 


This calculation is a little longer than the other, the division taking 
more time than the multiplication. 


3. Given a = 100, b = 49.5. To find 4, B, c. 

We may use either tan 4 = a/b or cot 4 = b/a to find 4. The 
latter gives the easier calculation. We have 

49 5 

cot 4 = TM = - 495a 

Hence, from the Tables, 4 = 63° 40'; and from (2), B = 26° 20'. 
To find c we may use either 




RIGHT TRIANGLES 


43 


c 2 = a 2 + b 2 or esc A = - 

a 

c = V a 2 + b 2 c = a esc A 

= Vl0,000 + 2450 = 100 X 1.116 

= VI2A50 = 111.6. 

= 111 . 6 . 

31. Checking a solution. Since errors are very likely to 
occur in solving a triangle, one should check the work. To do 
this, select a formula which has not already been used 
and which involves at least two of the parts of the triangle 
that were unknown. In this formula substitute the calcu¬ 
lated values. If the formula is verified, at least to a very 
close approximation, the solution checks; if not, there is 
probably an error in the work, and the solution should be 
gone over in an attempt to find the error. 

To check Example 1 of § 30 we select 

tan B = -, 
a ’ 

a formula which has not been used in the solution, and which in¬ 
volves all three unknowns. From the Table, 

tan B = tan 49° 40' = 1.178. 


By division, 


b 19.06 
a ~ 16.18 


1.178. 


The two results are the same; the solution checks. 

Another formula which could have been selected to check the 
solution is 

c 2 = a 2 + b 2 . , 


The calculation involved in using this formula is simplified by the 
aid of a Table of Squares, which we shall explain in the next section. 






44 


TRIGONOMETRY 


EXERCISES 

Solve the following triangles , and check your solution. In 
every case C =90°. The other two given parts are: 


I. A = 14° 20', c = 75. 

3. B = 26° 33', a = 25. 

5. B = 24° 21', h = 35. 

7. a = .23, h = .41. 

9. b = 621, c = 985. 

II. a = 3.03, c = 5.05. 

13. a = 55.12, h = 36.82. 
15. a = 3.684, c = 5.111. 
17. A = 77° 9', a = 654.3. 
19. A = 18° 8', h = 399. 


2. A = 38° 50', c = 4.5. 

4. B = 61° 27', a = 55. 

6. 5 = 78° 18', b = .48. 

8. a = 290, b = 150. 

10. b = .072, c = .123. 

12. a = 250, b — 350. 

14. a = 1.250, b = 2.500. 
16. a = 5.810, c = 7.952. 
18. A = 9° 27', a = 36.17. 
20. A = 83° 4', 6 = 36.7. 


32. Squares of numbers. In Table I at the end of the 
book we find the approximate values of the squares of num¬ 
bers from 1.00 to 9.99. Its use is illustrated in the follow¬ 
ing examples. 

Example 1. — To find (5.92) 2 . On page 3, go down the column 
headed N to 5.9, then across to the column headed 2. The ap¬ 
proximate value required is found to be 35.05. 

2. To find (5.925) 2 . We interpolate with the aid of the adjacent 
table (it should be done mentally 
after a little practice) and obtain 
the correction, 

x = 5/10 x 11 = 5.5 = 6 

approximately. We then have 
the approximation, 

(5.925)2 = 35.05 + .06 = 35.11. 

3. To find (59.25)2. We have 

59.25 = 10 X 5.925; 

(59.25)2 = 10 2 X (5.925)2 = 100 X 35.11, from Example 2, 

= 3511. 



N 

W 2 


f [5.920 

35.051 

10 

5 L.5.925 

J 


5.930 

35.16 







RIGHT TRIANGLES 


45 


Similarly, 

(592.5) 2 = KXT X (5.925) 2 = 351,100; 
(.5925) 2 - .3511; 

(.05925)2 = .003511. 


It should now be clear how the approximate value of the 
square of any number whatever is found. We may formulate 
the rule: For a given number, shift the decimal point to the 
right (or left ) k places to obtain a number between 1 and 10. 
Find the square of this from the Table. Shift the decimal 
point in this result 2 k places to the left (or right) to get the 
required square. 

33. Square roots. The square root of a number n in 
the interior of Table I is given by the corresponding number 
N read off from the left of the row and the top of the column 
in which n lies. We may, therefore, use the Table of Squares 
for the extraction of square roots. 


We note that the interior numbers lie between 1 and 100. We 
get the square roots of numbers in this range directly, though inter¬ 
polation may be needed. Thus 

V3496 = 1.870, V3L96 = 5.912. 


A number which does not lie between 1 and 100 can be expressed 
as the product of such a number by a power of 10 whose square root 
is simple. Thus 


349.6 = 100 X 3.496, 
3496. = 100 X 34.96, 
34960. = 10000 X 3.496, 

Hence 

V349J = VlbO X V3.496, 
= 10 X 1.870, 

= 18.70, 

V3496. = 59.12, 

V34960. = 187.0. 


.3496 = .01 X 34.96, 
.03496 = .01 X 3.496, 
.003496 = .0001 X 34.96. 

V^496 = VbT X V34.96, 
= .1 X 5.912, 

= .5912, 

V^3496 = .1870, 

V.003496 = .05912. 












46 


TRIGONOMETRY 


It should now be clear how the approximate square root 
of any number whatever can be found by use of the Table. 
A rule may be formulated as follows: For a given number 
shift the decimal point an even number of places, say 2 k, to 
the right (or left) to get a number between 1 and 100. Find the 
square root of this number from the Tables. In this square 
root shift the decimal point k places to the left (or right) to get 
the required number. 


EXERCISES 

Find the values of the squares of the following numbers to 
four places by use of the Table of Squares: 


1. 3.418; 

782.4; 

.06193; 

.2613. 

2. 4.872; 

51.32; 

.6666; 

.001818. 

3. 5.555; 

3892; 

.002468; 

.9876. 

4. 3.142; 

642.50; 

.02992; 

.3333. 

Find the square roots of the following 

numbers to four places 

I use of the Table of Squares: 


5. 6.742; 

38.18; 

.05932; 

.00342. 

6. 4.884; 

989.8; 

.004614; 

.01111. 

7. 3.333; 

7777; 

.05678; 

.217. 

8. 2.222; 

81.81; 

.9999; 

.00045. 

9. 3.629; 

48.19; 

574.2; 

.08765. 

10. 5.678; 

68.24; 

3693; 

.5791. 

★ 34. Approximations. 

Significant 

figures. In applica- 


tions of trigonometry we employ approximations to the 
exact values of lengths and angles; and, as we have already 
stated, the tabulated values of the functions are not exact. 
It is often of importance to know what accuracy we can 
expect from our calculations under these circumstances. 

In discussing this, let us first remark that the values given 
in the Tables are as nearly exact as they could be made by 


RIGHT TRIANGLES 


47 


using the given number of digits.* Hence when we find 
that sin 17° 20' = .2979 we may feel sure that the exact 
value lies between .29785 and .29795, or in other words, 
that the error is less than .00005. When a value is found 
by interpolation the error may be a little larger, but when 
the differences involved are small this error is in general less 
than 1 in the last place. Thus when we find by interpolation 
that sin 17° 23' = .2987, we can feel confident that this 
value is in error by less than .0001. 

It will be assumed in this book when we give a measure¬ 
ment of a length that the error is less than 1 in the last place 
where a digit other than zero occurs. Thus if we have given 
a = 3.12 m., we understand that the error is less than .01 m.; 
or if a = 3120 m., that it is less than 10 m. However, if one 
or more zeros are written after the decimal point at the end 
of a given value, they are to be considered significant, and 
the error is (presumably) less than 1 in the place of the 
final zero. For example, if we have a = 3.1200 m., it is to 
be inferred that the error is less than .0001 m. It is thus 
seen that 3.12 m. and 3.1200 m. have slightly different 
meanings. 

If a given number ends with one or more zeros that do 
not follow a decimal point, those digits are not to be con¬ 
sidered significant unless the contrary is stated. For ex¬ 
ample, if we are given a = 31200 m., we are to assume that 
the final zeros are not significant digits, and that the error 
is something less than 100 m. The same measure could have 
been expressed without final zeros as a = 31.2 km. If it is 
given that 11 a — 31200 m. to four significant figures,” the 
first zero is considered significant, and precisely the same 
thing would be meant by a = 31.20 km., — the error is less 
than 10 m., or 0.01 km. 

We may now define the significant figures (or digits) of a 
number as its digits beginning with the first that is not zero, 

* The digits are the numbers 0, 1, 2, . . . , 8, 9. 


48 


TRIGONOMETRY 


and ending in general with the last that is not zero. Ex¬ 
ceptional cases where final zeros are significant are those 
indicated in the preceding paragraphs. 

To turn from measurement of lengths to that of angles, 
we first consider an example. We find from the Tables that 

sin 14° = .2419, sin 15° = .2588. 

Thus a change of 1° in the angle corresponds to a change in 
the second figure of its sine. A glance at the Tables shows 
that in general the sine and cosine change about 1 in the 
second figure when the angle changes by 1°. It turns out, 
as might be inferred from considerations such as these, 
that the accuracy of the measurement of an angle to the 
nearest degree corresponds roughly to that of the measure¬ 
ment of a length to two significant figures. Measurements 
to the nearest 10' correspond roughly to three significant 
figures, to the nearest 1' to four, and to the nearest 5" to 
five significant figures in measurements of length. 

Consider now the accuracy of results obtained by calcula¬ 
tions with approximate values. Suppose, for illustration, 
that 

a = 316.2, b = 13.15, 

are correct to four significant figures. Then 
cl -{- b = 329.35; 

but there may be an error of nearly .1 in a, hence the final 
5 in a + b is not to be relied upon. Since the error in the 
value of a + b cannot be much greater than .1, we accept the 
3 in the tenths place as significant, but reject the final 5. In 
general, when two numbers are added, if their last significant 
figures occur in the same decimal place, the final significant 
figure of their sum occurs in that place. But if in one the 
final figure which is significant is in an earlier decimal place 
than in the other, the final significant figure in their sum 


RIGHT TRIANGLES 


49 


occurs in that earlier place. It should be noted that the 
error in a sum may be larger than the errors in the separate 
numbers, for the errors may accumulate. Thus the error 
in a sum may be larger than 1 in the final significant figure. 

The discussion for the difference of two approximate 
values is very similar to the preceding. 

In the case of multiplication the conclusions are somewhat 
different. Suppose 

a = 316.2, b = .15, 

then 

ab = 47.430. 

But how many of these figures are to be retained? Assuming 
only that a lies between 316.1 and 316.3, and b between .14 
and .16, we can conclude only that ab lies between 

.14 X 316.1 = 44.254 and .16 X 316.3 = 50.608. 

It is seen that only two figures of the product ab should be 
retained. Accordingly we write ab = 47 and recognize that 
the last digit may be in error. In general, the number of 
significant figures in a product should be the same as that 
of the factor having the fewer such figures; or if both factors 
have the same number p of such figures there should also be 
p in the product. Similar statements hold for a quotient. 
The error may be greater than 1 in the final place. 

In a computation requiring several operations the errors 
may accumulate to much more than 1 in the last significant 
figure, but as a rule errors tend to counteract each other and 
the final result is likely to have only a small error in that 
figure. 

A slightly greater accuracy is usually obtained in the 
computed value if at each step in the calculation we retain 
more figures than the preceding rules would allow. At the 
end of the computation, however, we should retain only as 


50 


TRIGONOMETRY 


many significant figures as those rules, if applied at each 
step, would permit. 

Example . — In a right triangle we have 

a = 4.27, c = 10.21, 

these values being approximate. Find A and b. 

We use the formulas 


sin A = -, b = Vc 2 - a 2 . 
c 7 


In calculating a/c, we retain four figures for slightly greater ac¬ 
curacy, although a has only three, and according to our rules a/c 
should have no more. We find 

sin A = .4182. 


Hence, to the appropriate number of significant figures, that is, to 
the nearest 10', A = 24° 40'. From the Tables, 


hence, 


c 2 = 104.2, a 2 = 18.23; 
b = Vc 2 - a 2 = V85.97 = 9.27. 


We have here retained only the justifiable number of significant 
figures for b. To check our work, we use the formula 

b = c cos A. 


We have 


c cos A = 10.21 X .9088 = 9.279 


to four figures. Since we had b = 9.27, the check is satisfactory. 


EXERCISES 

How many significant figures are there in each of the fol¬ 
lowing numbers considered as approximationst 

1. (a) 3817.2; (b) .00214; (c) 3.812 X 10 3 ; (d) 2.70 X 10~ 4 ; 
(e) 93,000,000. 

(a) 21.12; (b) .01010; (c) 2.0 X 10 2 ; (d) 2.777 X lO' 6 ; 
(e) 240,000. 


2. 





RIGHT TRIANGLES 


51 


3. With an ordinary foot-rule try to measure the length 
of a table to the nearest hundredth of an inch. Repeat the 
measurement four times. How large an error do you think 
is likely in your measurements? How many significant 
figures should you retain in your approximate value of the 
length? 

4. With an ordinary foot-rule try to measure the length 
of a page of this book to a hundredth of an inch. Repeat 
the measurement several times. Answer the questions of 
Exercise 3 for these measurements. 

5. If we have the measured values a = 36.2, b = 81.5, 
find limits between which the exact value of ab must lie. 
Similarly for a/b. 

6. Proceed as in Exercise 5 for a = 3.624, b = 81.5. 

Solve the following right triangles and retain the appropriate 
number of significant figures, assuming that the data are meas¬ 
urements: 

7 . A = 31° 20', c = 65.0. 

8. A — 59° to the nearest minute, and b = 41.00. 


35. Isosceles triangles. If in an isosceles triangle ABC, 
where AC = BC, we drop a perpendicular from C to AB, 
we get two equal right triangles c 

ADC and BDC. This fact may be 
used in solving an isosceles triangle. 

Example . — Solve the isosceles 
triangle, where the base is 21.25 ft. 
and the angle at the vertex is 37° 26'. 

In the triangle ADC we have 


D = 90 c 


AD 


2L25 

2 


y u 

/ D 


C 


Fig. 

44 

K 

- = 10.62, 



B 


ZACD = 


37° 26' 


18° 43'. 





52 


TRIGONOMETRY 


Then 

AC = AD esc Z.ACD, 

= 10.62 X 3.116 = 33.09 = BC, 

A = 90° - ZACD = 71° 17' = B. 

36. Regular polygons. If lines are drawn 
from the center of a regular polygon to its 
vertices, it is divided into equal isosceles 
triangles. If the polygon has n sides the 
angle at the vertex of each of these triangles 
is 360°/n. If a side AB, a radius AC, or an 
apothem CD is given, the other parts can be 
found by solving a right triangle. 

EXERCISES 

In the following Exercises retain the appropriate number of 
significant figures in each answer . The notation of Figure 44 
is used in Exercises 1 to 5. 

1. Given A = 50° 12', c = 4826. Find C, a, and b. 

2. Given C = 22° 46', c = 5164. Find A, a, and b. 

3. Given a = 3846, c = 2354. Find A, C, and b. 

4. Given A = 12° 16', a = 6891. Find C, c, and b. 

6. Given C = 88 ° 52', a = 8686 . Find A, c, and b. 

6. In a regular octagon, the length of a side is 2.32 in. 
Find the radius of the circumscribed circle and the apothem. 

7. In a regular hexagon, the apothem is 4.86 in. Find the 
perimeter. 

8. A regular decagon is inscribed in a circle whose radius 
is 10.00 in. Find the perimeter and the area of the decagon. 

37. Applications to heights and distances. Trigonometry 
undoubtedly had its origin in attempts to find certain angles, 
heights, and distances by indirect measurement. It is said 
that Thales of Miletus (about 600 b.c.) showed how to 
find the height of a pyramid, or the distance from the shore 





RIGHT TRIANGLES 


53 


to a ship at sea, by a method which is essentially that of 
trigonometry. 

At the present time surveyors and navigators make con¬ 
stant use of trigonometry in finding directions, distances, and 
heights. Let us see how a few such problems may be solved. 

Suppose a surveyor wishes to find the distance between two 
trees A and B on opposite sides of a stream. He can measure 
on one shore along a line perpendicular to a convenient 
distance AC (Fig. 46), measure the angle ACB, and find 



4 


'' Angle of 
elevation of C 


B 


Horizontal 
Fig. 47 


•H 


the required distance by solving the right triangle ACB. 

Suppose he wishes to find the distance from a position A 
to a flagpole BC of known height (Fig. 47) without leaving 
the position A. Assuming that A and B are in the same 
horizontal plane, and that BC is vertical, he may measure the 
angle BAC , which is called the angle of elevation of C for the 
observer at A, and solve the 
right triangle ABC for the re¬ 
quired distance AB. 

Suppose a navigator on board 
ship wishes to find how far he 
is from a certain rock R on shore 
at the water’s edge. If he sights 
with the appropriate instrument from A and observes that 
the line AR (Fig. 48) is depressed below the horizontal line 
AH by a certain amount, called the angle of depression of R 
as observed from A, and if he knows the height AH of his in- 


Horizontal 

•J Angle of 

^depression 
""^^of R 


R 


Fig. 48 












54 


TRIGONOMETRY 


strument above the water, he may solve the right triangle 
ABR and find the required distance. (We observe that the 
angle of depression of R for an observer at A equals the 
angle ARB, which is the angle of elevation of A for an ob¬ 
server at R.) 

At the end of this section we shall give a number of exer¬ 
cises more or less like those we have just presented. It 
will be helpful for the student to adopt the following method 
of procedure: 

(1) Read the problem carefully, then draw a figure to 
some convenient scale which will show those lines and angles 
which are given and those to be found. 

(2) Draw auxiliary lines if necessary, and decide on the 
simplest plan for solving the problem. 

(3) Write down all necessary formulas. 

(4) Carry out the numerical calculations, retaining the 
appropriate number of significant figures in each answer. 

(5) Check the results. 


EXERCISES 

1. At a point 256 ft. from a flagpole, and on a level with 
the base, the angle of elevation of the top is 18° 20'. How 
tall is the pole? 

2. A stick 10.5 ft. long stands vertically and casts a shadow 
12.8 ft. long in a horizontal plane. What is the angle of 
elevation of the sun? 

3. A sailor at sea level observes that the angle of eleva¬ 
tion of the top of a rock 290 ft. high is 22°. How far is he 
from the top of the rock? How far from the point at sea 
level directly under the top of the rock? 

4. A boy observes that the angle of elevation of his kite 
is 35° when 220 yd. of string are out. Assuming that the 
string is straight, how high is the kite? 

5. From the deck of a boat 45 ft. above water level the 


RIGHT TRIANGLES 


55 


angle of depression of a stone on the beach, at the water’s 
edge, is 5°. How far away is the stone from the observer? 

6. From a window ledge almost exactly 40 ft. above a 
level street the angle of depression of the base of a building 
across the street is 21°, and the angle of elevation of its top 
is 62°. Find the height of the building. 

7. Two points A and B are on opposite sides of a lake. A 
line from A to C running due West is 392.2 yd. long. A line 
from B to C running due South is 521.4 yd. long. How far 
is it from A to B ? 

8. To find the distance across a river from A to B, a sur¬ 
veyor ran a line along one shore from A to C perpendicular 
to AB and of length 350 ft. He measured the angle ACB; 
it was 52° 30'. Find the width AB. 

9. A navigator sailed a course of 211° (see p. 6) for 2 hr. 
25 min. at 22.2 mi. per hr. Assuming that the surface of 
the water was a plane, how far South and how far West was 
his final position from his initial position? 

10. One port is 61 mi. East and 37 mi. South of another. 
What is the direction (or course) from the first to the second 
port? Assume that the surface of the earth is a plane. 

11. Two observers at A and B in a horizontal plane ob¬ 
serve a captive balloon C. The points A, B, and C lie in a 
vertical plane, with C above a point between A and B. 
The distance AB is 1570 yd. At A the angle of elevation of 
C is 25° 20', at B it is 34° 30'. How high is the balloon above 
the plane of the observers? 

12. From a ship running on a course N 5° E along a shore 
the bearing of a rock is observed to be N 32° E. After run¬ 
ning 350 yd. the bearing of the rock is N 51° E. If the ship 
continues oil its course, how close will it come to the rock? 

13. The angle of elevation of the top of a spire from a 
point A in a horizontal plane is 22° 23'; from a point B 
100.0 ft. nearer it in the same plane the angle of elevation is 
35° 12'. How high is the top of the spire above the plane? 


56 


TRIGONOMETRY 


14. A tunnel into the earth descends at an angle of de¬ 
pression of 14°. When a man has descended 350 ft. along 
the tunnel how far is he below the level of his starting point? 

15. The planet Venus goes around the sun in an orbit 
which is practically circular, its distance from the sun being 
about 67 X 10 6 mi. The earth’s orbit is also nearly circular, 
the distance from the earth to the sun being about 93 X 10 6 
mi. What is the maximum value for the angle between the 
line from the earth to the sun, and the line from the earth 
to Venus? Will Venus ever be seen in the East in the 
evening? 

16. A surveyor starts at a point A, goes N 18° E 782 ft. 
to B, then S 47° E 691 ft. to C, then S 11° W 388 ft. to D. 
Find the direction and distance from D to the starting 
point A. 

17. A sailor Sails a course of 63° 20' for 21.37 mi. from A 
to B, then a course of 192° 50' for 31.21 mi. from B to C. 
Find the bearing and distance of A from C. 


CHAPTER III 


OBLIQUE TRIANGLES* 

38. General statement. In the preceding chapter we 
saw how a right triangle is solved. Let us now consider 
oblique triangles. 

In the first place, we may ask how many of the six parts 
(three sides, a, b, c, and three angles, A, B, C) of a triangle 
need to be given in order to determine the triangle. If we 
recall certain propositions of plane geometry we shall re¬ 
member that it is possible to construct a triangle when 
three of its parts are given in each of the following cases: 

Case I. When one side and two angles are given. 

Case II. When two sides and an angle opposite one of 
them is given (there is a possible ambiguity in this case; 
see § 44). 

Case III. When two sides and the included angle are 
given. 

Case iV. When three sides are given. 

It thus appears that any three parts of a triangle, provided 
they are not all angles, determine the triangle. It will be 
found desirable to take up solutions under each of these 
four cases separately. 

In order to solve a triangle we need relations among the 
parts in the form of equations from which we can obtain the 
value of each unknown part in terms of those that are given. 
It turns out that the following relations are sufficient for 
the purpose: 

* In a brief course it may be desirable to omit all of this chapter except 
§§ 38-41; in this case these sections should be deferred until Chapter VIII 
has been completed. 


57 


58 


TRIGONOMETRY 


(1) The formula A + B + C = 180°. 

(2) The law of sines (§ 40). 

(3) The law of cosines (§ 41). 

We shall find that another cosine formula is convenient for 
checking computations. 

Some of the calculations, when a high degree of accuracy 
is required, are tedious on account of the amount of arith¬ 
metic involved. In a later chapter (Chapter IX) we shall 
take up simplifications that are made possible through the 
use of logarithms. 

39. Sine and cosine of obtuse angles. The Tables give 
values of the trigonometric functions for acute angles only. 

Since an obtuse angle 
may occur in an oblique 
triangle, we must see 
how values of the func¬ 
tions of such angles can 
be found. 

Let 6 be an obtuse 
angle; then 180° — 6, 
its supplement, is acute. 
Referring to the defini¬ 
tions of § 15, we draw Figure 49. We choose P and P' so that 
r = r'. 


Y 


P 

P' 


-A? / 

y \ 

^r<?8O°-0 r 

M x O 

X’ M'X 

Fig. 

49 


It is not difficult to show that the triangles OMP and OM'P' 

are congruent. Hence, taking due account of the signs of 

each quantity, we have, 

y — MP — M'P' = y', 

x = OM = -OM* = -x'. 

We therefore have 

sin 0 = - = ^ = sin (180° - 0), 
r r 

cos 0 = - = — %■ = —cos (180° — 0). 
r r 




OBLIQUE TRIANGLES 


59 


Thus, the sine of an obtuse angle equals the sine of the supple¬ 
mentary angle (which is acute); and the cosine of an obtuse 
angle is the negative of the cosine of the supplementary angle. 

As examples, we have 

sin 121° 12' = sin 58° 48' = .8554, 
cos 121° 12' = -cos 58° 48' = -.5180. 

40. The law of sines. The formula known by this name 
is derived as follows. 

In a triangle ABC let a , b, c be the sides opposite the 
angles A, B } C , respectively. From the vertex C drop a 


c c c 



Fig. 50 


perpendicular upon the side AB (produced if necessary), 
calling the foot of the perpendicular D. Then we have 

. A DC . D DC 

sin A = —r- , sm B - - 

b a 


It is to be noted (Fig. 50) that these equations hold whether 
the angles A and B are both acute, or A is acute and B 
obtuse, or A obtuse and B acute. The student may draw 
figures in which either A or B is a right angle and verify 
that the formulas still are true. 

On dividing these equations member by member, we 
obtain, with a change of order, 


( 1 ) 


a _ sin A 
b sin B * 


Since any two sides of a given triangle may be called a 
and b, the formula may be stated in words thus: Any two 









60 


TRIGONOMETRY 


sides of a triangle are to each other as the sines of the opposite 
angles. This is known as the law of sines. 

For a given lettering of the triangle, it follows that we 
have, in addition to equation (1), 

a _ sin A b _ sin B 

c sin C’ c sin C 

The last three equations are equivalent to each of the fol¬ 
lowing continued equations: 

a _ b _ c 
sin A sin B sin C 

( 2 ) 

sin A _ sin B _ sin C 
a ~ b c 

EXERCISES 

Find the numerical values of the following functions by use 
of the Tables: 

1. sin 102° 20'; sin 168° 14'. 

2. sin 121° 30'; sin 175° 12'. 

3. cos 98° 50'; cos 155° 17'. 

4. cos 112° 30'; cos 167° 11'. 

Find all possible values of the angle A, acute or obtuse , 
which satisfy each of the following equations 5 to 8: 


5. 

sin A = 

.9088; 

sin A = .4362. 

6. 

sin A = 

.4041; 

sin A = .9055. 

7. 

cos A = 

.8689; 

cos A = —.5997. 

8. 

cos A = 

.9407; 

cos A = —.8270. 


9. Draw the appropriate figure for the proof of the law 
of sines for the case A = 90 p , and verify the formula. 

10. Proceed as in Exercise 9 for the case B = 90°. 

11. Prove from figures that 

a sin A 
c sin C 










OBLIQUE TRIANGLES 


61 


41. The law of cosines. This extension to oblique tri¬ 
angles of the Pythagorean theorem expresses any side, a, in 
terms of the other sides b and c and the opposite angle A. 
As a formula it is written 

(1) a 2 = b 2 + c 2 — 2 be cos A. 

We shall give two proofs. In the first we employ the 
methods of elementary geometry; in the second the methods 
of coordinate geometry. 

First method . We use Figure 50. We have in every case, 
from the right triangles BDC and ADC, 

a 2 = DC 2 + DB 2 , DC 2 = b 2 — AD 2 , 

and hence 

(2) a 2 = b 2 -AD 2 A- DB 2 . 

In the first two triangles of Figure 50, where the angle A 
is acute, we have respectively 

DB = c-TD , L)B = AD-c. 

In either case 

T)B 2 = c 2 - 2 cAD + AD 2 . 

We see from the triangles that in either case 
AD = b cos A. 

On substituting these last two equations in (2) and simpli¬ 
fying we have formula (1). 

In the third triangle of Figure 50, where the angle A of the 
triangle ABC is obtuse, we observe that 

DB = c + AD, 

whence 

DB 2 = c 2 + 2 cAD + AD 2 . 

We also have 

AD — b cos Z DAC. 


62 


TRIGONOMETRY 


The substitution of these last two equations in (2) gives on 
simplification 

a 2 = b 2 _i_ C 2 2 be cos ZDAC. 

Since ZDAC is the supplement of the angle A of the given 
triangle ABC, we have from § 38, 

cos A = —cos ZDAC, 

and hence the preceding equation is equivalent to formula 
(!)• 

The student maj" readily verify that the formula (1) is true 
when A = 90° or B = 90°. It will then have been proven 
for all cases. 

Second method. We take A as the origin of a system of 
rectangular coordinates, the positive x-axis extending along 



AB (Fig. 51). Let the coordinates of the vertex C be 
(x,y). Then in every case we have 

a 2 = y 2 + DB 2 , 
y 2 — b 2 — x 2 , 

and hence _ 

a 2 = b 2 - x 2 + DB\ 

When we give due regard to signs we have in every case 
DB = c — x. 

Substituting this in the preceding equation we get 
a 2 = b 2 + c 2 — 2 cx. 









OBLIQUE TRIANGLES 


63 


And since in every case 

x = b cos A 

the formula (1) follows at once. 

Since a was any side of the triangle, it follows that formu¬ 
las similar to (1) hold when the letters are changed. We 
thus have 

(3) b 2 = a 2 + c 2 — 2 ac cos B, 

(4) c 2 = a 2 + b 2 — 2 ab cos C. 

*42. Another cosine formula. From Figures 51 of § 41, 

we find that in every case, when due regard is paid to signs, 

AD = b cos A , DB = a cos B, 

c = AD + DB. 

Hence, in every case, 

(1) c = b cos A + a cos B. 

Similarly, 

(2) b = a cos C + c cos A, 

(3) a = b cos C + c cos B. 

EXERCISES 

1. Draw the appropriate figure and prove formula (1), 
§ 41, in case B = 90°. 

2. Proceed as in Exercise 1 in case A = 90°. Also in 
case C = 90°. 

3. Show that in case B = 90°, each of the three formulas 

(1), (3), (4) of § 41 is equivalent to the formula a 2 = b 2 — c 2 . 

4. Draw the appropriate figures and prove formula (3), 

§41. 

5. Proceed as in Exercise 4 for formula (4), § 41. 

6. Proceed as in Exercise 4 for formula (2), § 42. 

7. Prove the law of cosines (equation (1), § 41) from 


64 


TRIGONOMETRY 


equations (1), (2), (3) of §42, by multiplying them re¬ 
spectively by —c, —6, and a, then adding and simplifying. 

8. Prove equation (3), § 41, by a method similar to that 
suggested in Exercise 7. 


*43. Case I. Given two angles and one side. An ex¬ 
ample will suffice to indicate how any problem coming 
under this case is solved. 

Example. — Given a = 262, A — 36° 20', B — 75 50 . 
To find C, b, c. 


We draw Figure 52, letting 1 cm. 



formula containing that unknown 
law of sines, written in the form 


represent 100 units, and estimate 
therefrom b = 430, c =410, 
C = 70°. 

In the numerical calculation 
of the unknowns we determine 
the unknown angle from the 
formula A + B + C = 180°, 
from which we have 

(1) C = 180° — (A + B). 

To find the side b, we need a 
but no other. We see that the 


b _ sin B 
a ~ sin A ’ 


will suffice. Solving for the unknown we have 
_ a sin B 

( 2 ) 6 = inrx' 

Similarly, to find c we have 

£ _i s ^ n C 
a _ sin A’ 


and hence 


a sin C 
sin A 


(3) 


c 





OBLIQUE TRIANGLES 


65 


As a check we may use the formula 

(4) a = c cos B + b cos C , 


which contains the three parts which were unknown. 

On substituting the given values in these solution-formulas (1), 
(2) and (3), we have 


C = 180° - (36° 20' + 75° 50') = 180° - 112° 10' = 67° 50', 


262 sin 75° 50' 
sin 36° 20' 


262 

.5925 


X .9696 = 442.2 X .9696 = 428.8, 


262 sin 67° 50' 
C “ sin 36° 20' 


262 

.5925 


X .9261 = 442.2 X .9261 = 409.5. 


Our calculated values check roughly with the estimated values 
found from the figure. To get a more accurate check we substitute 
our values in the right-hand member of (4). We have 

c cos B + b cos C = 409.5 cos 75° 50' + 428.8 cos 67° 50' 

= (409.5 X .2447) + (428.8 X .3773) 

= 100.2 + 161.8 = 262.0. 


Since we had given a = 262, the check is excellent. 

If the given values are exact, the use of four-place values found 
from the Tables gives us four significant figures in the answer. But 
if the given values for this problem are merely approximate, then 
only three figures in our results are retained as significant, since 
each term of the calculation has that accuracy. Our results should 
then be written 

C = 67° 50', b = 429, c = 410. 


EXERCISES 

Solve the following triangles , and check your answers. Give 
results to four significant figures: 

1. A = 32°, C = 67°, b = 120. 

2. B = 46°, C = 65°, a = 3.5. 

3. A = 15°, B = 33°, a = 25. 

4. A = 112°, C = 18°, c = 6.6. 

6. B = 66° 20', C = 71° 10', b = 12.5. 






66 


TRIGONOMETRY 


6. A = 52° 30', B = 82° 50', b = 75.5. 

7 . A = 22° 40', B = 131° 50', a = .824. 

8. B = 100° 10', C = 45° 40', c = 6120. 

9. A = 44° 44', C = 66° 22', c = 51.67. 

10. B = 101° 13', C = 41° 27', b = .02183. 

★ 44. Case II. Given two sides and the angle opposite 
one of them. Geometrical discussion. Suppose the given 
parts of the triangle are A, a and b. To construct the tri¬ 
angle (Figs. 53-56) we first draw the angle A, and lay off 
the length b on one side, locating the vertex C. To locate 
the vertex B, we draw a circle K with a as radius and C as 
center. The vertex B must lie on this circle and on the 
second side of the angle A. At this step we find that there 
are several possibilities, which we shall take up in succession. 

First , suppose that the angle A is Qcute. Let D be the foot 
of the perpendicular from C to the second side of the angle A. 
The length of CD is b sin A. We have four sub-cases: 

(1) If the given side a is shorter than CD the circle K 
does not intersect the second side of the angle A (Fig. 53), 
and there can be no triangle with the given parts. 



Fig. 53 


C 



(2) If a = CD, the circle is tangent to AD at D (Fig. 53), 
and the right triangle ADC is the required triangle. 

(3) If the side a is longer than CD but shorter than b, 
the circle K cuts AD at two points Bi and B 2 (Fig. 54), 
either of which may be the third vertex; hence there are two 
triangles, ABiC and AB 2 C, which have the given parts 





OBLIQUE TRIANGLES 


67 


A, a and b. We note that the angle B 2 of the one triangle, 
AB 2 C, is the supplement of the angle Bi of the other triangle, 
ABiC. 

(4) If the side a is at least as long as b (Fig. 55), the 
circle K cuts AD in only one point B on the side AD of the 
angle A, and hence one and only one triangle is possible. 



A > 90 c 
Fig. 56 


Fig. 55 


Second, suppose that the angle A is a right angle or obtuse . 


Then 


(1) If the side a is not longer than b (Fig. 56), there is no 
triangle. 

(2) If the side a is longer than b (Fig. 56), there is exactly 
one triangle. 

Hence in Case II there may be no solution, one solution, or 
two solutions. We note that the unknown angle B opposite 
the known side b is acute when there is just one solution; 
but that there are two angles, one acute, the other obtuse, 
supplements of each other, when there are two solutions. 

For solving a triangle which comes under Case II it is 
desirable to construct a figure first, at least roughly, to see 
whether there will be no triangle, one triangle, or two tri¬ 
angles. 

Because there is a possibility of two triangles, Case II 
is sometimes called the ambiguous case. 

Trigonometrical solution. Suppose a, b, and A are given. 
To find the angle B, we may use the law of sines in the form 


sin B _ sin A 
b a 





68 


TRIGONOMETRY 


We have 

(i) 


sin B 


b sin A 
a 


If a < b sin A, we see that sin B > 1, which is impossible. 
Hence there is no solution. 

If a = b sin A, we have sin B = 1; hence B = 90°. The 
problem may be solved as one in right triangles. 

If a > b sin A, we have sin B < 1, and B may have either 
of two values — an acute angle Bi which is given in the 
Tables, or its supplement B 2 (see § 39). We write down 
both angles and proceed on the assumption that two tri¬ 
angles are possible — a triangle ABiC and a triangle AB 2 C. 
The same method is used for the solution of each. If the 
angles at C in the two triangles are Ci and C 2 respectively, 
we have 

(2) Ci = 180° - (A + £i), C 2 = 180° - (A + B 2 ). 

It may happen that A + B 2 > 180°, in which case C 2 is an 
impossible angle for a triangle and there can be only one 
triangle, ABiC. The side ci is determined from the relation 

Ci _ sin Ci 
a sin A 


whence 

(3) 


a sin Ci 
sin A 


If the second triangle exists, we find c 2 by the similar 
formula > 


(4) 


_ a sin C 2 
sin A 


The solutions are checked by the relations 
a = b cos Ci + ci cos B i, 
a = b cos C 2 + c 2 cos B 2) 






OBLIQUE TRIANGLES 


69 


respectively. It is noted that the check formulas have not 
previously been used in the solution, and that they relate 
all three of the computed parts. 

Examples. — 1. Given a = 25, b = 33, A = 44°. To 
find c, B, and C. 

By construction, letting 1 cm. represent 10 units, we find that 
there are two triangles ABiC and AB 2 C. Let ci, B h C x be the 
unknown parts of the first triangle, 
and c 2 , B 2 , C 2 those of the second 
triangle. 

Our estimates by measurements are: 
ci = 33, B x = 67°, Ci = 70°; 

c 2 = 15 , B 2 = 111°, C 2 = 24°. 

The equations to be used in solving 
are (1), (2), (3), and (4). We have 
first 

b sin A 33 X .6947 22.925 



sin B 


= .9170. 


a 25 25 

Hence 

B = 66° 29' or 180° - 66° 29' = 113° 31'. 
The first angle is B x , the second B 2 ; 

B x = 66° 29', B 2 = 113° 31'. 

Solving the triangle AB X C, we have 

Ci = 180° - (44° + 66° 29') = 69° 31'; 

then 

a sin Ci 25 X .9368 


.6947 


= 33.71. 


To check, we find 

a = b cos Ci + Ci cos B X = (33 X .3499) + (33.71 X .3990) = 25.00; 

since a — 25, the check is excellent. 

Solving the triangle ABzCi, we have 

C 2 = 180° - (44° + 113° 31') = 22° 29' 









70 


TRIGONOMETRY 


and 


a sin C 2 25 X .3824 
= sin A = .6947 


13.76. 


To check, we have 

b cos C 2 + c 2 cos B 2 = (33 X .9240) +(13.76 X —.3990) = 25.005, 

which agrees well with the given value a = 25. To find cos B 2 we 
used the relation cos 113° 31' = —cos 66° 29' (see § 39). 

If the given values are regarded as exact, the calculations, in 
which approximate values to four significant figures are used, give 
results with that number of significant figures. But if the data are 
regarded as values given by measurements our answers should 
be written 


B ! = 66°, Ox = 70°, cx = 34; 

B 2 = 114°, C 2 = 22°, c 2 = 14. 

2. Given a = 33, b = 25, A = 136°. To find B, C, and c. 

In this example we shall illustrate only one step of the solution. 
From the equation 


sin B 


b sin A 
a 


25 X .6947 
33 


= .5263, 


we find 

Rx = 31° 45', B 2 = 180° - 31° 45' = 148° 15'. 

Then 

Ci = 180° - (A + Bi) = 180° - 167° 45' = 12° 15', 
C 2 = 180° - (A + B 2 ) = 180° - 284° 15', impossible. 


There is therefore only one solution for this example. 

3. Given a = 22,9, b = 33, A = 44. To find B, C, and c. 

The construction in this case would leave one in doubt as to the 
number of solutions. We have 

33 X .6947 22.93 

sm B - 22 9 - 2 2.9 


which is greater than 1. Since there is no angle whose sine is 








OBLIQUE TRIANGLES 


71 


greater than 1, there is no triangle having the given parts. We 
have the case a < b sin A, illustrated in Figure 53. 

EXERCISES 

Construct a figure for each of the following sets of data, tell 
how many triangles are possible, and estimate the values of the 


unknown parts: 

1. A = 30°, 

a = 40, 

b = 100. 

2. 

A 

= 60°, 

a = 60, 

b = 100. 

3. 

A 

= 30°, 

a = 50, 

b = 100. 

4. 

A 

= 60°, 

a = 87, 

b = 100. 

5. 

A 

= 30°, 

a = 60, 

b = 100. 

6. 

A 

= 60°, 

a = 95, 

b = 100. 

7. 

A 

o ~ 
O 
CO 

II 

a = 120, 

b = 100. 

8. 

A 

= 60°, 

a = 150, 

b = 100. 

9. 

A 

= 120°, 

a = 60, 

b = 100. 

10. 

A 

= 150°, 

a = 70, 

b = 100. 


11 . A = 120°, a = 120, b = 100. 

12 . A = 150°, a = 150, b = 100. 

Solve the following triangles, having given: 

13. B = 50°, b = 36, c = 55. 

14. B = 75°, b = 80, a = 78. 

15. C = 13°, b = 62, c = 45. 

16 . C = 62°, b = 10.0, c = 75. 

17. C = 125°, b = 1.25, c = 2.36. 

18. A = 140°, c = 2.57, a = 2.18. 

19. A = 34° 21', a = 3.007, b = 4.153. 

20. A = 66° 43', a = 518.0, b = 612.9. 

*45. Case III. Given two sides and the included angle. 

We shall give two methods for solving a triangle which 
comes under this case. The first is convenient if no great 
accuracy is desired, and especially if only the third side is 
required, not the two unknown angles. The second is 


72 


TRIGONOMETRY 


shorter when great accuracy is desired, and all unknown 
parts are to be found. 

First method. An example will suffice to make the method 
clear. Suppose we are given 6 = 15, c = 21,A=35°. 

In constructing a figure let a length of 1 cm. represent 10 units. 

We estimate the unknowns as follows: 
a = 12, B = 47°, c = 99°. 

To compute a we may use the law of co¬ 
sines, § 41, 

a 2 = b 2 + c 2 — 2 be cos A, 

since a is the only unknown part in this formula. The use of a 
Table of Squares simplifies the calculation. The angle B may be 
found from another form of the law of cosines, 



whence 


&2 _ a 2 C 2 _ 2 ac cos B, 


cos B = 


a 2 + c 2 - b 2 
2 ac 


Finally, we have C = 180° - (A + B). We may check by the 
law of sines, written in a form containing the computed side a and 
the last angle found, which was C. We write it, for simplicity of 
calculation, 

a sin C — c sin A. 


The computation follows: 


i* = 225 + 441 - 630 (.8192) = 149.9 


cos B 


149.9 + 441 - 225 
514.1 


.7117 B 


.*. a = 12.24. 
= 44° 42'. 


C = 180° — (A + B) = 180° - (79° 42') = 100° 18'. 


For the check we have 

a sin C = 12.24 X .9839 = 12.04 
c sin A = 21 X .5736 = 12.05. 


Second method (by right triangles). If A, b, and c are the 
given parts, we drop a perpendicular CD from C to the side 





OBLIQUE TRIANGLES 


73 


AB (produced if necessary). In the right triangle ADC 
thus obtained, we solve for AD and DC. In the right tri¬ 
angle BDC we then have DC, and BD is easily found. We 
may therefore solve this triangle for the side a and the angle 
DBC. In case D falls outside of B on A B produced (see the 
third of Fig. 59) the required angle B of the triangle ABC is 
the supplement of the angle DBC; in other cases it equals 


C C c 



Fig. 59 


that angle. The angle C is found from the relation that the 
sum of A, B, and C is 180°. 

In the example worked out by the first method we would use the 
formulas (Fig. 58) 

DC = b sin A AD = b cos A 

DC 

DB = c — AD tan B = jjjj 

a = or a* = DC + DB 

sm B 

C = 180° - (A + B) 

and the check a sin C = c sin A. 

Having b = 15, c = 21, A = 35°, we find 

DC = 15 X .5736 = 8.604 
AD - 15 X .8192 = 12.29 
DB = 21 — 12.29 = 8.71 

tan B = = .9878 B = 44° 39' 

a = v'74.03 + 75.86 - V149.89 = 12.24 
C = 180° - 79° 39' = 100 ° 21'. 

For the check we have 

a sin C = 12.24 X .9838 = 12.04 
c sin A = 21 X .5736 = 12.05, 







74 


TRIGONOMETRY 


If our data were approximate measurements we could abbreviate 
our calculations by using only three significant figures, and avoiding 
interpolations. The results would then be written 
B = 45°, . a - 12, C = 100°. 

*46. Case IV. Given three sides. Triangles coming 
under this case can always be solved by the law of cosines. 
One form of this, 


cos A = 


b 2 + c 2 


2 be 


enables us to compute the angle A. Likewise from 

a 2 + c 2 — b 2 


cos B = 
cos C = 


2 ac 
a 2 + b 2 - 
2 ab 


we may compute B and C. As a check we may use 
A + B + C = 180°. 

Example. — Given a = 51, b = 65, c = 20. 

We construct a figure, letting 1 cm. represent 20 units, and esti¬ 
mate the angles: A = 38°; B = 126°; C = 14°. The calculation 
follows: 


38° 53' 



= 2601 

2 ab 

= 6630 

Z> 2 = 4225 

2 be 

= 2600 

c 2 = 400 

2 ac 

= 2040 

2024 

.7785 


cos ^ = 2600 = 

.-.A = 

_ -1224 



cos B - 2040 

= -.6000 .*. 

„ 6426 



cos C = 6630 = 

.9692 

.*. C = 


14° 16' 


Check: A + B + C = 180° 1'. 


If only two significant figures are desired in the answers, we can 
shorten the work by using only three significant figures in the 
calculations, and by omitting interpolations. 







OBLIQUE TRIANGLES 


75 


It is obvious that if the sides are given to five or more 
significant figures and corresponding accuracy is required 
in the angles, the calculation will be very long. In Chapter 
IX we shall give a shorter computation by use of logarithms. 

EXERCISES 

In each of the following triangles find the unknown side, 
having given: 

1. a = 84, c = 72, B = 69°. 

2. a = 67, b = 81, C = 58°. 

3. 6 = 63.2, c = 18.4, A = 122° 30'. 

4. a = 189, c = 524, B = 132° 40'. 

5. a = 26.12, b = 31.72, C = 132° 52'. 

6.6 = 38.15, c = 71.10, A = 121° 34'. 

In each of the following triangles find the two unknown 
angles, having given: 

7. 6 = 362, c = 471, A = 58° 30'. 

8. a = .182, c = .261, B = 112° 20'. 

Solve and check the following triangles, having given: 

9. 6 = 28, c = 47, A = 29°. 

10. c = 28, 6 = 47, A = 151°. 

11 . 6 = 48.2, c = 61.9, A = 102° 10'. 

12. 6 = .501, c = .236, A = 61° 20'. 

13. a = 36, 6 = 46, c = 56. 

14. a = 7.4, 6 = 6.2, c = 4.1. 

16. a = 581, 6 = 781, c = 1081. 

16. a = 409, 6 = 236, c = 295. 

17. a = 576, 6 = 817, c = 311. 

18. a = 8.247, 6 = 7.631, c = 6.848. 

19. a = 363.4, 6= 317.2, c= 491.6. 

20. A = 28° 4', 6 = 88.71, c = 63.48. 

21. a = .2413, B = 121° 12', c = .8124. 

22 . a = 6.819, 6 = 5.241, C = 158° 27'. 


76 


TRIGONOMETRY 


MISCELLANEOUS EXERCISES 


In the following problems the student should note the implied 
accuracy of measurements and retain the appropriate number 
of significant figures (p. 46) in the results. 

1. From a ship a lighthouse had a bearing (p. 6) of 123°; 
after the ship had gone due East 1.3 mi., the lighthouse had 
a bearing of 158°. Find the distance from the ship to the 
lighthouse in each position. 

2. An observer on board a ship notes the bearing of a rock 
to be 26° 30'. After traveling due North 750 ft., he finds 
the bearing to be 45° 00'. If he continues on the course 
how close will he get to the rock? 

3. A surveyor running a line due East from A encounters 
a swamp which he must go around. He wishes to continue 

the line on the other side of the 
swamp. At a point B on his 
line he changes his direction to 
N 36° 00' E for 335 yd., to C, 
then turns to S 57° 00' E. 

Fig- 61 How far should he continue on 

this course to reach a point D on the continuation of AB? 



How far is D from B ? 

4. Two circles whose radii are 27 in. and 32 in. intersect. 
The angle between the tangents at a point of intersection 
is 37°. Find the distance between the centers. 

5. To find the distance between 
two points A and C which are 
separated by an impassable bar¬ 
rier, a man measures a line from 
A to B of length 120 yd., then 
from B to C of length 95 yd. If 
the angle CAB is 45°, how far is it from A to C? 

6 . Two sides and a diagonal of a parallelogram are of 
lengths 34 in., 22 in., and 17 in., respectively. Find the 
angles at the vertices of the parallelogram. 



Fig. 62 


OBLIQUE TRIANGLES 


77 


7. To find the distance between two inaccessible points 
P and Q, a line AB lying in a plane with PQ and the angles 
a, a', of Figure 63 are measured. Find PQ if 

AB = 525 yd. « = 55° 20', a! = 102° 10', 

0 = 48° 30', 0' = 97° 50'. 


8. To find the length h of a line PQ, a distance AB = d 
is measured on a line AP perpendicular to PQ ; and the angles 
a and 0 (Fig. 64) are observed. Let the distance BP = x. 



Fig. 63 


Q 



Fig. 64 


Show that h and x are given by the formulas 

_ d d tan 0 

h= - j x = -t— 

cot a — cot 0 cot a — cot (3 

{Hint. Write down equations for cot a and cot 0, and 
solve for h and x.) 

9. Show that if a = 21°, and 0 = 32°, the formula for h 
in Exercise 8 becomes h— d. If a. = 26° 30', what value of 
0 makes h = d? For these latter values of a and 0, what 
is the value of x ? A navigator who is traveling a course 
AB can easily measure the angle corresponding to a at any 
time and the distance d traveled between two observations. 
How could he use these results if he wishes to know how far 
abeam (distance PQ) he will pass a rock Q if he continues 
his course AB? 

10 . If the height of a statue on top of a building is 15 ft., 
and at an unknown distance m from the foot of the building 
in a horizontal line the angles of elevations of the top and 










78 


TRIGONOMETRY 


bottom of the statue are 40° and 32° respectively, what is the 
value of m? 

11 . In Figure 65, the angles a and p are 
measured. If m is also known, show that 
h is given by the formula 

h — m (tan P — tan a ). 

12. In Figure 66, the point P is above a 
horizontal plane ABC, PC being vertical. 

The line AB is measured, AB — a; 
and the angles a, a', P, P', are ob¬ 
served. Show that the height h 
of P above the plane ABC is 

a sin |3 tan a' a sin a tan P' 
Fig. 66 sin ( a + 0) sin ( a + P) 




h = 


13. If in Exercise 12 there is a balloon at P, and if 
a = 4500 ft., a= 30°, p= 75°, a' = 40°, how high is the 
balloon? What should P' be in this case? 

14. The earth and the planet Venus move around the sun 
in orbits which are approximately circles with the sun at 
the center, the radii being 92,800,000 mi. and 66,800,000 mi. 
respectively. When an astronomer observes the angle be¬ 
tween the line from the earth to the sun and the line from 
the earth to Venus to be 27° 40', how far is Venus from the 
earth? 








CHAPTER IV 


REDUCTION FORMULAS. LINE VALUES. GRAPHS 

Trigonometric tables enable us to find the values of func¬ 
tions of acute angles. We now consider the problem of 
reducing a function of an angle that is not acute to a function 
of an angle that is given in the Tables. A first simplifica¬ 
tion is effected in certain cases by adding to or subtracting 
from the given angle a multiple of 360°; according to the last 
paragraph of § 15 (p. 18) the functions of the new angle are 
the same as those of the old. Thus we have 

sin 735° = sin (735° - 720°) = sin 15°, 
tan (-190°) = tan (-190° + 360°) = tan 170°. 

It remains to develop formulas which will, for example, prove 
that tan 170° is equal to —tan 10°. We shall find that such 
reduction formulas are valid even when the reduced angle is 
not acute. 

When we have thus obtained formulas that enable us to 
compute the values of functions of any angle, we shall find it 
useful to represent the functions graphically. This will be 
accomplished by means of figures employing line values , and 
by graphs in rectangular coordinates. 

47. Functions of 180° — 0. An angle between 90° and 
180° can always be expressed as 180° - 9 , where 9 is a suitably 
chosen acute angle. We now develop formulas which ex¬ 
press each of the six functions of 180° — 9 in terms of func¬ 
tions of 9. In the case of the sine and cosine these formulas 
are closely related to those of § 39 (p. 58). 

In Figure 67a the length OP' = r' on the terminal side of 
the angle 180° - 9 is taken equal to OP = r on the terminal 

79 


80 


TRIGONOMETRY 


side of angle 0. The right triangles OMP and OM'P' will 
then be equal, since their angles at 0 are equal and we have 
OP = OP' by construction. It follows that each side of one 
triangle is of the same length as the corresponding side of 



Fig. 67a 


Y 



Y 




the other, but when we interpret this statement in terms of 
coordinates we must take account of plus and minus signs. 
While r' and r are both positive, and y' and y are of the 
same sign, x' and x are of opposite sign. Thus we have 

(1) x' = -x, y' = y, r' = r. 

These equations, together with the definitions of the trigono¬ 
metric functions, give the following identities: 

sin (180° — 0) = — = ^ = sin 6; 

cos (180° —6) = ~j — ~~ = —cos 6; 


(2) 
















REDUCTION FORMULAS. LINE VALUES 


81 


tan (180° — 0) = ^ = — = -tan 0; 

x —x ’ 

cot (180° - 0) = - r = — = -cot 0; 

( 2 ) y , v 

sec (180° — 0) = r — = — = —sec 0; 

x x 

esc (180° — 0) = = - = esc 0. 

V V 

The preceding relations hold also when 0 is an angle 
terminating in the second, third, or fourth quadrants, as 
illustrated in Figures 67b, 67c, 67d. An inspection of each 
case will show that equations (1) are always true, and that 
the identities (2) are therefore still valid. 

Examples. — 1. Find the value of tan ( — 237°). 

By adding 360° to —237° we obtain the angle 123°, whose func¬ 
tions are the same as those of —237°. We then express 123° as 
180° — 57° and use the identity for tan (180° — 0), substituting 
0 = 57°. Thus we have 

tan (-237°) = tan (123°) = tan (180° -57°) = -tan 57°. 

2. Find an angle 0 terminating in the second quadrant 
and such that cos 0 = —0.5736. 

We first use the tables to find the acute angle a such that cos a = 
0.5736; the value of <* is 55°. From the second of identities (2), 

cos (180° — a) = —cos a = —0.5736 

so that a solution of our problem is 

0 = 180° - « = 125°. 

We shall see later (p. 93) that there can be no other solution between 
90° and 180°. 

48. Functions of 180° + 0. An angle between 180° and 
270° can be expressed in the form 180° + 0, where 0 is an 
acute angle. In Figure 68a we take the angle XOP equal to 0 


82 


TRIGONOMETRY 


and XOP' equal to 180° + 0, with OP' equal to OP. The 
right triangles OM'P' and OMP are equal; hence 
x' = -x, y' = -y, r' = r. 

It follows that 

sin (180° + 0) = jj = = ~ sin e > 

cos (180° + 0) = p = = -cos 0, 

tan (180° + 0) =-' = — = - = tan 0. 

X X X 




We prove similarly the formulas 

cot (180° + 0) = cot 0, 
sec (180° + 0) = —sec 6 , 
esc (180° + 0) = —esc 0. 

From Figure 68b, where 6 terminates in the second quad¬ 
rant, the same equations and identities could be deduced. 
They are also true when 6 is an angle terminating in the 
third or the fourth quadrant. Thus the six identities just 
obtained hold true for all angles 0. 

49. Functions of 360° — 0 and of —0. According to a 
statement made at the beginning of this chapter, the func¬ 
tions of 360° — 6 are the same as those of —6. 










REDUCTION FORMULAS. LINE VALUES 


83 


Any angle between —90° and 0° can be expressed as — 0, 
where 0 is a positive acute angle. In Figure 69a, the angle 
XOP is equal to 0, and XOP' is -0. We take OP' = OP, 



1 

p 



y 



M 

o 

\rd) x 1 

M' 


r\. 

y' 


1 

o r 

Fig. 69a 




so that triangles OM'P' and OMP are equal, and 


x' = x, 

It follows that 

sin (—0) 

cos (—0) 
tan ( — 0) 


y' = -y, r' = r. 


t 

r' 

x' 


r' 



—y • „ 

—- = —sin 6, 
r 

X 

- — cos 0, 
r 

—- = — tan 6. 

x 


Similarly 

cot (—6) = —cot 0, 
sec (—0) = sec 0, 
esc (—0) = —esc 0. 


Using Figure 69b for an angle 0 terminating in the second 
quadrant, and additional figures for angles 0 terminating in 
the third and fourth quadrants, the student should prove 
that the preceding identities are true for all angles 0. 

50. General rule for n • 180° d= 0. By means of the 
formulas of the three preceding sections we can reduce a 
function of an angle 540° ±0 = 3- 180° ± 0 to a function 









84 


TRIGONOMETRY 


of 9 by subtracting 360° from the angle and using an identity 
of § 47 or § 48. Similarly, — 180° ± 6 = — 1 • 180° zb 9 may 
be treated by adding 360°. Functions of —360° zb 9 = 
-2 • 180° ± 9 reduce to those of zb 9. By such means we 
can express functions of n • 180° zb 9, where n is zero or any 
positive or negative integer, in terms of functions of 9. 
The results are summarized in the following working rule: 

Any given function of an angle n • 180° zb 9 is equal either 
(i a ) to the same function of 9, or else ( b ) to the negative of that 
function: 

Given function of (n • 180° zb 0) = =b same function of 0. 

The + sign is to he taken on the right side of this formula if, 
when 9 is acute, the angle n • 180° zb 9 terminates in a quadrant 
for which the given function of that angle is positive; the — sign 
if the given function of that angle is negative when 9 is acute. 

Examples. — 1. Prove that cos ( — 1176°) = —cos 84°. 

The angle —1176° can be written as —7 • 180° + 84°; hence 
cos ( — 1176°) is equal either (a) to +cos 84° or (b) to —cos 84°. 
Since —1176° terminates in the third quadrant its cosine is nega¬ 
tive, hence statement (b) is the correct one. We could also have 
started by adding 4 • 360° to —1176°. 

2. Find the value of sin ( — 137°). 

Since sin ( — 137°) = sin ( — 180° +43°), and since —137° ter¬ 
minates in the third quadrant, we have 

sin (-137°) = -sin 43° = -.6820. 

3. Find an angle 9 terminating in the fourth quadrant 
and such that tan 9 = — 2. 

We first find the acute angle a such that tan a = 2. By inter¬ 
polation we obtain a = 63° 26'. Since 

tan (360° — a) = —tan a = — 2, 
it follows that one solution is 

6 = 360° - « = 316° 34'. 

Any angle differing from this by a multiple of 360° is also a solution. 


REDUCTION FORMULAS. LINE VALUES 85 
EXERCISES 

1. By reference to the rule of § 50, prove the following 
relations: 

(a) sin 123° = sin 57°; (c) tan 325° = tan 145°; 

(b) cos (-123°) = -cos 57°; (d) cot 500° = -cot 40°. 

Reduce each expression in the following Exercises 2 to 5 to 
a function of an acute angle , using the rule of § 50: 

2. (a) sin 150°; (b) cos 235°; (c) tan 320°; 

(d) cos (-20°); (e) cot (-140°); (f) esc (-230°). 

3 . (a) tan 170°; (b) cos 215°; (c) sin 280°; 

(d) tan (-35°); (e) sec (-140°); (f) cot (-325°). 

4 . (a) cos 459°; (b) tan 117° 38'; (c) sin 316° 21'; 

(d) cot 1039°20'; (e) sec (-700°); (f) esc 582° 28'. 

5. (a) cos 128° 23'; (b) cot 342° 15'; (c) sin 714°; 

(d) sec 1280°13'; (e) tan (-1000°); (f) esc 478° 43'. 

6. By means of the Tables, find the value of each ex¬ 
pression in Exercise 4. 

7. By means of the Tables, find the value of each ex¬ 
pression in Exercise 5. 

8. Find an angle 0 terminating in the second quadrant and 
such that sin 0 = .3090. 

9. Find an angle 0 terminating in the second quadrant and 
such that cos 0 = —.9205. 

10. Find an angle 0 terminating in the fourth quadrant 
and such that tan 0 = —.6100. 

11 . Find an angle terminating in the fourth quadrant and 
such that cos 0 = .3821. 

12. Find the angles 0 terminating in the third quadrant 
and such that cot 0 = .9192. 

13 . Find the angles 0 terminating in the third quadrant 
and such that sin 0 = —.7287. 

14 . Find the rectangular coordinates of the points whose 

polar coordinates are: (a) (10, 120°); (b) (2, 225°); 

(c) (h -35°); (d) (5, 143° 22'). 


86 


TRIGONOMETRY 


15. Find the rectangular coordinates of the points whose 

polar coordinates are: (a) (1, 240°); (b) (5, 135 ); 

(c) (20, -136°); (d) (.3, 327° 14'). 

16. Find the polar coordinates of the points whose rec¬ 
tangular coordinates are: (a) ( — 1, 1); (b) (3, —3); 

(c) (-4, -1); (d) (-5, 7). 

17. Find the polar coordinates of the points whose rec¬ 
tangular coordinates are: (a) ( — 1); (b) ( — 2,—2\ / 3); 

(c) (-3, 10); (d) (3.3, -4.8). 

18. By reference to the rules of § 50, prove the formulas: 

(a) sin (0 - 360°) = sin 0; (c) tan (540° - 0) = -tan 0; 

(b) sin (i9 -180°) = - sin 0; (d) cos (■-180° - 0) = - cos 0. 

19. Construct figures to illustrate § 48 for cases where 0 
terminates in the third and fourth quadrants, and prove for 
these cases the formulas for cot (180° + 0), sec (180° + 0), 
esc (180° + 0). 

20. Construct the additional figures suggested in § 49 and 
prove the formulas cot ( —0) = — cot 0, sec ( —0) = sec 0, 
esc ( — 0) = —esc 0, for the corresponding cases. 

51. Functions of 90° ±0. In § 27 (p. 36) we have shown 
that each function of an acute angle 0 is equal to the corre¬ 
sponding cofunction of the complementary angle 90° - 0. 
Figure 70a illustrates a proof similar to those of the preceding 
sections. In this figure OP = OP' by construction, and the 
angle MOP is equal to the angle M'P'O. It follows that 

x' = y, y f = x, r' = r, 

and sin (90° -9) = p = ~ = cos 6, 

cos (90° - 9) = p = V - = sin 0, 

tan (90° — 9) =—, = -= cot 9. 

x y 




REDUCTION FORMULAS. LINE VALUES 


87 


Similarly, 

cot (90° 
sec (90° 
esc (90° 



0) = tan 0, 
0 ) = esc 0, 
0) = sec 0. 


Y 



Figure 70b indicates how to show that the preceding 
identities hold also for angles 0 that terminate in the second 
quadrant. The formulas are, in fact, true for all angles 0. 


* p 


■x 


Fig. 71a 


Y 




For functions of 90°+ 0, Figures 71a and 71b illustrate cases 
where 0 terminates in the first and fourth quadrants respec- 

















88 


TRIGONOMETRY 


tively. As in the preceding sections we take OP = OP'; the 
triangles OMP and OM'P' are then equal, with angle MOP 
equal to angle M'P'O. It follows that 

x' = —y, y' = x, r' = r. 

We readily deduce the identities 

sin (90° + 0) = cos 0, cos (90° + 0) = —sin 0, 

tan (90° + 0) — —cot 0, cot (90° + 0) = —tan 0, 

sec (90° + 0) = — esc 0, esc (90° + 0) = sec 0. 

52. Functions of 270° ± 0. In Figures 72 and 73 we 

illustrate only cases where 9 is acute. The identities that 
follow are true for all angles 9, no matter in what quadrant 



they terminate. We take OP = OP' and observe that in 
all cases angle MOP is equal to angle M'P'O . 

For the angle 270° — 9 (Fig. 72) we have 

x> = —y, y' = -X, r' = r, 
and from these relations we conclude that 

sin (270° — 9) = — cos 0, cos (270° — 0) = — sin 0, 

tan (270° - 0) = cot 0, cot (270° - 0) = tan 9, 

sec (270° - 0) = -esc 9, esc (270° - 0) = -sec 0. 

Similarly, for 270° + 0 (Fig. 73) we have 

x' = y, y' = -x , r' = r, 










REDUCTION FORMULAS. LINE VALUES 


89 


sin (270° + 9) = —cos 9, cos (270° + 9) = sin 9, 

tan (270° + 9) = -cot 9, cot (270° + 9) = -tan 9, 

sec (270° + 9) = esc 9, esc (270° + 9) = -sec 9. 

53. General rule for n • 90° db 9, where n is odd. In 

§50 we can replace n • 180° ± 0 by n • 90° ± 9 provided 
n in this last expression is restricted to be zero or a 
positive or negative even number. Sections 51 and 52 
yield a corresponding rule for n • 90° ± 9 where n is odd. 
When n is equal to 1 or 3 the preceding sections give the 
results directly, while cases where n has other positive or 
negative odd integral values reduce to those where n = 1 or 
3 if we add or subtract suitable multiples of 360°. 

Our working rule is: 

Any given function of an angle n • 90° d= 9, where n is odd, 
is equal either (a) to the corresponding cofunction of 9, or else 
(b) to the negative of that cofunction: 

Given function of (n • 90° ± 0) = d= cofunction of 0 (n odd). 

The + sign is to he taken on the right side of this formula if, 
when 9 is acute, the angle n • 90° ± 9 terminates in a quad¬ 
rant for which the given function of that angle is positive; the 
— sign if the given function of that angle is negative when 9 is 
acute. 

Example. — Express cos ( — 500°) in terms of a function 
of an acute angle. 

The angle -500° is equal to -5*90°- 50°, hence its cosine is 
equal either to sin 50° or -sin 50°. Since -500° terminates in 
the third quadrant its cosine is negative, hence cos (-500°) = 
—sin 50°. We could also have proceeded as follows: 

cos (-500°) = cos (-500° + 720°) = cos 220° 

= cos (270° - 50°) = -sin 50°, 


cos (-500°) = cos (-3 • 180° + 40°) = -cos 40° 
= -sin 50°. 


or 


90 


TRIGONOMETRY 


EXERCISES 

1. By reference to the rule of § 53, prove the following 
relations: 

(a) cos 115° = —sin 25°; (c) cot (—40°) = —tan 50°; 

(b) sin 460° = cos 10°; (d) sec (—1000°) = esc 10°. 

2. Express as a function of an acute angle, using the rule 
of § 53, each function of an angle in Exercise 2 (p. 85). 

3. Express as a function of an acute angle, using the rule 
of § 53, each function of an angle in Exercise 3 (p. 85). 

4. Solve Exercise 14 (p. 85), using the rule of § 53. 

5. Solve Exercise 15 (p. 86), using the rule of § 53. 

6. Prove the following relations, using the rule of § 53: 

(a) sin ( — 90° — 0) = — cos 

(b) tan (0 — 90°) = — cot 0; 

(c) cos (-270° - 0) = sin6; 

(d) cot (0 - 270°) = -tan0. 

7. Prove the first six formulas of § 51, using figures where 
6 terminates in the third and fourth quadrants. 

8. Prove the last six formulas of § 51, using figures where 
6 terminates in the second and third quadrants. 

54. Line values. We now describe the construction of a 
figure in which the value of each function of an angle 0 is 
given by the length of a directed line-segment. These seg¬ 
ments are called the line values of the functions. Such a 
representation is more convenient for certain purposes 
than the ratio definitions (§ 15, p. 17). 

First draw the familiar figure in which 0 is the angle XOP 
and the triangle OMP has the side MP perpendicular to the 
z-axis; take OP so that it is one unit long. Draw a circle 
about 0 with radius OP, which we shall call the unit circle. 
Let it intersect the positive £-axis at A, and the positive 
2 /-axis at B (Fig. 74). It follows that OA = OB = 1. At 
A and B draw tangents to the circle intersecting OP (pro¬ 
longed) in points T and T' respectively. According to the 


REDUCTION FORMULAS. LINE VALUES 


91 


usual conventions regarding directed line-segments, one 
along a horizontal line and directed to the right is positive 
and one directed to the left is negative, while one pointing 



vertically upward is positive and one pointing downward 
is negative. It has been agreed (§ 8, p. 7) that on OP seg¬ 
ments having the direction OP are positive and those having 
the opposite direction are negative. 

For an acute angle 9, as shown in Figure 74a, we have, 


( 1 ) 

( 2 ) 


sin 0 = 


MP MP 


OP 


1 


= MP, 


OM OM 

cos e = op = — = OM, 


also, since triangles OMP, OAT, and OBT' are similar, 
MP AT AT _ 

OM OA 1 
OM BT' BT' 


(3) 

(4) 

(5) 

( 6 ) 


tan 0 


cot 8 MP OB 1 BT ’ 
. OP _ OT _ 0T_ _ 
sec 6 “ OM OA 1 0T ’ 


CSC 0 = 


OP OT' OT' 


MP OB 


ttf; = OT'. 


The line segment indicated at the extreme right of each of 
the above formulas is called the line value of the correspond¬ 
ing function. 










92 


TRIGONOMETRY 


All the equations we have written above are also true 
when 0 terminates in the second, third, or fourth quadrants 
(Fig. 74b, 74c, and 74d), the lengths of the segments being 
taken positive or negative as we have already indicated. 
To prove this statement we first note that the equations for 
sin 0 and cos 0 hold in all cases, by our definitions of these 
functions. For the other four functions, their expressions 



as ratios of the sides of the triangle OMP also follow defini¬ 
tions previously made. From the similarity of triangles 
OMP, OAT, and OBT' our equations remain true except, 
possibly, that negative signs might need to be introduced. 
That this is not the case may be verified by inspecting the 
figures, which show, for example, that in each case when 
tan 0 is positive the same is true for A T, and when tan 0 is 
negative, A T is negative. The student should prove that a 
similar statement is true for each of the other functions. 

In all cases, therefore, equations (1) to (6) give the line 
values of the six trigonometric functions. 

55. Variation of sin 0 and tan 0. By means of line values 
we may easily note how the value of a function changes when 
the angle increases. Thus Figures 74 make it evident that 
when OP is rotated about 0 from the position OA to the 
position OB, that is, when the angle 0 increases from 0° to 90°, 










REDUCTION FORMULAS. LINE VALUES 


93 


MP = sin 9 increases steadily from the value 0 to the value 1. 
Similarly, when 9 increases from 90° to 180°, sin 9 remains 
positive but decreases steadily from 1 to 0; when 9 increases 
from 180° to 270°, sin 9 is negative and decreases steadily 
from 0 to —1; and when 9 increases from 270° to 360°, 
sin 9 is negative and increases from —1 to 0. 

We could proceed similarly with each of the other functions 
but it will be sufficient to state the facts for tan 9 = AT. 
As 6 increases from 0° the point T rises, and AT can be made 
as large as we please by taking 9 sufficiently near 90°. This 
is equivalent to the statements that tan 9 is positive and 
increases steadily as 9 increases from 0° toward 90°, and that 
tan 9 becomes infinite as 9 approaches 90°. Similarly, by 
taking account of changes in A T we see that when 9 increases 
from 90° to 180°, tan 9 is negative and steadily increases 
from extremely large negative values to zero. A brief way 
of indicating these facts is to say that tan 9 increases from 
0 to +oo (read “infinity”) as 9 increases from 0° to 90° 
and that it increases from — oo to 0 as 9 increases from 90° to 
180°. It is also true that tan 9 increases from 0 to +oo as 9 
increases from 180° to 270°, and from — oo to 0 as 9 increases 
from 270° to 360°. 

From the fact that when 9 increases from its value at the 
beginning of a quadrant to its value at the end of that quad¬ 
rant, each trigonometric function of 9 either increases stead¬ 
ily or else decreases steadily, we infer that no trigonometric 
function can have the same value for two different angles ter¬ 
minating in the same quadrant unless these angles differ by a 
multiple of 360°. 

We can further conclude by following the variation of the 
functions that there are at most two angles between 0° and 360° 
for which a function has a given value. Thus the equation 
sin 9 — a, where a is a positive number between 0 and 1, 
is satisfied by one value of 9 between 0° and 90°, by one value 
between 90° and 180°, and by no other value between 0° and 


94 


TRIGONOMETRY 


360°. If a were between 0 and — 1 there would be one solu¬ 
tion for 0 between 180° and 270°, one between 270° and 360°, 
and no others between 0° and 360°. The equation tan 0 = a 
has two solutions between 0° and 360° for every number a, 
positive or negative. If a is positive there is one solution 
0 between 0 and 90°, one between 180° and 270°, and no 
others between 0° and 360°; if a is negative there is one 
solution between 90° and 180°, one between 270° and 360°, 
and no others between 0° and 360°. 


EXERCISES 

1. Indicate which of the line values are positive and 
which are negative in Figures 74b, 74c, 74d, and thus verify 
the statement that each correctly represents the corre¬ 
sponding function of 0. 

Describe the variation of the following functions: 

2. cos 9. 3. cot 9. 

4. sec 9. 5. esc 9. 

By using line values , with suitable figures , prove the fol¬ 
lowing identities: 

6. sin (180° — 9) = sin0. 7. tan (90° + 9) = —cot 9. 

8. cos (180° + 9) = —cos 9. 9. sec (90° — 9) = csc0. 


Find all the values of 9 between 0° and 360° that satisfy the 
following equations. Use tables where necessary. 


10. tan 9 = • 

V3 

12. sin 0 = 1. 

14. sec 0 = — 2. 

16. sin 0 = —.2025. 

18. tan 0 = -2.4378. 


11. COS 0 = —r=- • 

V 2 

13. tan0 = 0. 

15. cot 0 = 2. 

17. cos 0 = .8297. 
19. esc 0 = 2.5300. 


* 56. Graphs in rectangular coordinates. In this section 
we recall the method of representing the variation of a 


REDUCTION FORMULAS. LINE VALUES 


95 


quantity by a graph in algebra. In the following section 
we shall apply this method to trigonometric functions. 

Consider, for example, the algebraic function 2 x — 3. 
We form the equation y = 2 x — 3, give to x various values, 
and compute the corresponding values of y. Thus if x = 0, 
we have y — 2*0 — 3=—3; likewise if x = 1, we have 

y = 2 • 1 — 3 = — 1. We tabulate a number of these pairs 

of values below. 

X \ -4: I -3 I -2 I -1 I 0 I 1 I 1.5 I 2 I 4 

y | -II | -9 | -7 C -5 | -3 | -1 1 0 | 1 | 5 


When the points (—4, —11), (—3, —9) and the others given 
by the preceding table are plotted, it is seen that they all 
lie on a straight line, as shown in Figure 75. This straight 
line is called the graph of the algebraic 
function 2 x — 3. 

Among the many purposes that are 
served by graphs we note two. First, 
by measuring coordinates of points on 
the graph we can find approximate 
values of the function for given values 
of x without any algebraic computa¬ 
tion. Thus for x = 1.3 we could find 
the value of 2 x - 3 by setting a ruler 
so that its edge is perpendicular to the 
X-axis at x = 1.3 and measuring the 
distance from the z-axis to the point 
where the ruler’s edge intersects the graph. In the second 
place, the graph shows how the function increases or decreases 
as x increases. Thus the graph of 2 x — 3 shows that this 
function always increases when x increases, that we can give 
2 x — 3 a negative value which is numerically as large as we 
please by assigning to z a sufficiently large negative value, 
and that we can give 2 x - 3 as large a positive value as we 
please by assigning to x a sufficiently large positive value. 



Fig. 75 




































96 


TRIGONOMETRY 


Sometimes this is put more briefly by saying that 2 x — 3 
increases steadily from — oo to + oo when x increases from 
— OO to + CO . 

As another example we consider the graph of 1 — x 2 . 
We form the equation y = 1 - x 2 , give a succession of values 
to x, and compute the corresponding values of y. Thus for 
x = 2, we have y = 1 - 2 2 = -3. We give a table of 
values below: 

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 

y | —8 | —3 | 0 | 1 | 0 | -3 | -8' 

When the points (-3, -8), (-2, -3), etc., have been plot¬ 
ted, a smooth curve is drawn through them, as shown in 
Figure 76, and this we call the graph of 
1 - x 2 . 

From this graph we see that the algebraic 
function 1 — x 2 increases steadily from 
— oo to 1 as x increases from — oo to 0, and 
that it decreases from 1 to — oo as x in¬ 
creases from 0 to + oo. 

* 57. Graphs of the trigonometric func¬ 
tions. In order to obtain a graph of the 
function sin x we first represent angles 
by points on the z-axis, as shown in Figure 77. In the 
equation y = sin x, accordingly, we give to x a succession of 
values and compute y. In the following table corresponding 
values of x and y are shown: 

x | 0° | 10° | 20° | 30° | 60° | 90° 

y | 0 | .1736 | .3420 | .5000 | .8660 | 1.000 

If x is a negative angle or is greater than 90° we use the re¬ 
duction formulas of § 47 to § 53 (p. 89) in finding the values 
of y. When our table has been sufficiently extended and we 
have plotted the corresponding points, we draw a smooth 
curve through them and obtain the graph of sin x (Fig. 77). 


Y 













/ 

s 







n 


L 








\ 







\ 




1 





\ 



7 








t 





\ 



r 





\ 


L 








Fig. 76 



























REDUCTION FORMULAS. LINE VALUES 


97 


From this graph we read off results already noted re¬ 
garding the variation of sin x. Thus when x increases from 
0° to 90°, the ordinate of the graph, which gives the value 
of sin x, increases from 0 to 1. The student may similarly 
trace the further variation of sin x. 

It will be noted that the curve in Figure 77 is composed 
of arches above and below the z-axis which are alternately 




Fig. 78 


symmetrical and congruent. The graph also shows that 
the value of y at x = a d= 360° is equal to its value at x = a, 
where a is any angle whatsoever. This property of sin x is 












































































































































98 


TRIGONOMETRY 


expressed by saying that it has the period 360°. The graph 
repeats itself at intervals of 360°. 

In Figure 78 we show a graph of tan x, from which we easily 
trace the variation of tan x and note that this function has 
the period 180°. This graph clearly indicates the behavior 
of tan x when x approaches 90° or 270°. 

* 

EXERCISES 

Draw graphs of the following functions and discuss the 
variation of the functions by means of the figures: 

1. cos x. 2. cot x. 3. sec x. 4. esc x. 

5. Draw the graph of sin x — cos x. Is there a period? 

6. Draw the graph of sin x + cos x. Is there a period? 

7. Show by means of the graph of sin x that an equation 
sin x = a has either no solution or else infinitely many solu¬ 
tions. Show also by means of the graph that if a is numer¬ 
ically less than 1, the equation sin x = a has two and only 
two solutions in the interval 0° ^ x < 360°, and that both 
of these are in the interval 0° < x < 180° if a is positive. 

8. Discuss in the manner indicated in Exercise 7 the 
equation tan 6 = a, where a is any number (positive nega¬ 
tive, or zero). 


CHAPTER V 


FUNDAMENTAL IDENTITIES 

58. Trigonometric identities. In algebra an equation in 
one or more unknowns is called an identity if it holds for all 
values of the unknowns. Similarly an equation in terms of 
trigonometric functions of one or more angles is an identity 
if it holds when the angle or angles take on all possible values. 
By the phrase “all possible values” we mean all values except 
those for which a function in the identity is undefined,* or a 
denominator is zero. 

The reduction formulas of the preceding chapter are ex¬ 
amples of relations between trigonometric functions of an 
angle 0 and of a related angle which are true for all values of 
the angle 9 for which the functions are defined. In this 
chapter we shall first consider a still simpler class of identities 
involving functions of a single angle 9. We shall next de¬ 
velop the addition formulas which express functions of the 
sum and of the- difference of two angles in terms of func¬ 
tions of those angles. Further identities will be deduced 
as corollaries of the addition formulas. 

59. Formulas involving one angle. From the definitions 
of the six trigonometric functions in terms of each other and 
of x, y, and r (§ 15, p. 17), certain identities are immediately 
deduced. For example, in the section just referred to, the 
functions cot 9, sec 9, and esc 9 are defined as reciprocals of 


, cos 6 , and sin 0 respectively: 


C0t 9 = tan0 ’ 

sec 0 = “a j 
cos 0 

CSC 0 = 

sm 0 

tan0= c^0’ 

1 

COS 0 = - A , 

sec 0 

• 0 1 

sm 0 =- a 

CSC 0 


* It will be recalled that tan 90° and esc 180°, for example, have no meaning. 

99 




100 


TRIGONOMETRY 


Two more identities express tan 0 and cot 0 in terms of sin 0 
and cos 0. The first arises from the relation 


tan 0 


y 

y _ r _ sin 0 
x x cos 0 
r 


The other comes from the expression of cot 0 as the recipro¬ 
cal of tan 0. These identities are 


( 2 ) 


tan 0 = 


sin 0 
cos 0 ’ 


cot 0 = 


cos 0 
sin 0 


Another set of identities consists of corollaries of the law 
of right triangles which states that the square of the hypot¬ 
enuse is equal to the sum of the squares of the other two sides. 

In the x, y, r triangle (see Fig. 21, p. 17) the square of the 
base is x 2 , whether x is positive or negative; for in the latter 
case the length of the base is -x, and its square is {-x) 2 = x\ 
Similarly, the square of the altitude is y 2 , and the square of 
the hypotenuse is r 2 . We have, then, in all cases, 
x 2 “f" 2/2 = r 2 . 

Let us divide this identity through by r 2 and interpret the 
result in terms of trigonometric functions. This gives us 



(sin 0) 2 + (cos 0) 2 = 1. 

It is customary to write this 

sin 2 0 + cos 2 0 = 1. 

Similarly, if we divide by x 2 we have 



1 + tan 2 0 = sec 2 0. 



FUNDAMENTAL IDENTITIES 


101 


If we divide by y 2 , changing the order of terms, 



1 + cot 2 0 = esc 2 0. 


We have thus obtained the three identities 


(3) 


sin 2 0 + cos 2 0 = 1, 

1 + tan 2 0 = sec 2 0, 
1 + cot 2 0 = esc 2 0. 


The formulas of groups (1), (2), and (3) are of great 
importance in trigonometry, and must be memorized. 

*60. Formulas expressing the functions in terms of a 
single function. The identities of the preceding section 
furnish another method of solving such problems as that 
of Example 3 of § 22 (p. 30) where it was required to express 
all of the trigonometric functions in terms of sin 0. In order 
to solve the problem just referred to, we note that the first 
identity of group (3) can be solved for cos 0 as follows: 


cos 2 0 = 1 — sin 2 0, 


cos 0 = =b V 1 — sin 2 0. 


The ambiguous sign before the radical here indicates that 
there are two possible solutions for cos 0, of which only one 
is correct for a given angle 0. If 0 terminates in either the 
first or the fourth quadrant the function cos 0 is positive, 
while for the other quadrants it is negative. In the former 
case we have 

sin 0 = sin 0, cos 0 = + Vl - sin 2 0, 


OJ.JJL 1/ 

tan0 =-- 


sin 0 sin 0 


cos 0 -f Vl — sin 2 0 ’ 



+ Vl — sin 2 0 
sin 0 


sec 0 = 


1 


cos 0 -(- V1 — sin 2 0 * 










102 


TRIGONOMETRY 


If 9 terminates in the second or the third quadrant, the only 
change to be made in the above formulas is to replace the 
+ sign before each radical by a — sign. 

The problem of expressing all functions in terms of the 
cosecant is solved by replacing sin 9 by its equal 1/csc 9 
wherever the former occurs on the right of the formulas 
given in the last paragraph. The procedure for obtaining 
expressions in terms of cos 6 and sec 9 is similar to that which 
we have just indicated for sin 9 and esc 9. If we can solve 
the similar problem for tan 9, the modification for cot 9 is 
obvious. Let us, therefore, examine this remaining case, 
that of expressing the functions in terms of tan 9. From the 
second of the identities (3) we have 

sec 9 = =L Vl + tan 2 9, 

where the + sign is taken if 9 terminates in the first or the 
fourth quadrant, and the — sign if in either of the other 
quadrants. We then have 

„ 1 1 

cos 9 = -- = - , .. — • 

sec 9 iVl + tan 2 9 

The first of identities (2), written in the form 
sin 9 = cos 9 tan 9 , 

now gives us 

. . , tan 9 

sin 9 = cos 9 tan 9 = - , — . 

d= v 1 + tan 2 9 

The remaining functions, cot 9 and esc 9, are reciprocals of 
tan 9 and of the preceding expression for sin 9 respectively. 

The formulas obtained in this section give a new way also 
for solving such problems as those of Examples 1 and 2 of 
§ 22 (p. 29), where we are to find the values of all functions 
of 9 when the value of one is given. A better method is per¬ 
haps to use the identities of the preceding section directly. 







FUNDAMENTAL IDENTITIES 


103 


Thus if tan 0 = 5/12, as in Example 2 of § 22, we have 

sec 2 0 = 1 + tan 2 0 = 1 + ( T \) 2 = fff, 
sec 0 = =Lfj§, 

COS 0 = -- = ~~i~ 

sec 0 13 ’ 


sin 0 
cot 0 


cos 0 tan 0 = =h|| Xn = =t T 5 B -, 


tan 0 


esc 0 


sin 0 


= ±¥, 


where the + sign is to be retained if 0 is acute, and the — 
sign if 0 terminates in the third quadrant. 

+ 61. Simplification of expressions involving trigonometric 
functions. From the foregoing section it is evident that an 
expression involving one or more trigonometric functions 
can be transformed into an expression in terms of any single 
function. It is often advantageous to choose this last 
function so as to avoid the introduction of radicals. Thus 
the transformation 

sin 0 _ sin 0 _ 1 

1 — cos 2 0 sin 2 0 sin 0 

avoids radicals, while one is introduced in the following, 
with the attendant disadvantage of an ambiguous sign: 

sin 0 d= V1 — cos 2 0 _ 1 

1 — cos 2 0 1 — cos 2 0 ±Vl — cos 2 0* 

Some expressions, such as sin 0 + cos 0, cannot be given 
in terms of a single function of 0 without radicals, but it is 
to be noted that we can express each trigonometric function 
rationally in terms of any two that are not reciprocals of 
each other. Thus, if we choose these two as sin 0 and cos 0, 
their quotients are equal to tan 0 and cot 0, and their recipro¬ 
cals are sec 0 and esc 0. It is evident, therefore, that an 










104 


TRIGONOMETRY 


expression in three or more functions can be reduced to one 
in no more than two functions without introducing radicals 
that were not originally present. 

62. Proofs of identities. From the formulas of § 59 an 
unlimited number of identities can be deduced. A set is 
given at the end of this section, and the student is required 
to prove them as exercises. We will illustrate three methods 
of procedure. 


(a) We can transform one side of the identity into the other by 
means of algebraic processes and the formulas of § 59. Thus, to 
prove 


(1) 

we could write 


sin 9 

1 — cos 2 9 


= esc 0 


sin 9 sin 6 1 

1 — cos 2 9 ~ sin 2 9 ~ sin 9 ~ CSC 6 ' 

(b) We can transform both sides into one expression, 
prove (1) by the relations expressed in parallel columns: 


Thus we 


sin 9 

1 — cos 2 9 


esc 9 


sin 9 
sin 2 9 

1 

sin 9 


1 

sin 9 


(c) By working with the identity as a whole we may reduce it to 
one in which the expression on one side coincides with that on the 
other, or we may reduce the identity to one of the formulas of § 59. 
Thus (1) is an identity provided it is true that 

sin 9 = (1 — cos 2 9) esc 9, 


sin 9 = sin 2 9 


sin 9 


which is true if 






FUNDAMENTAL IDENTITIES 


105 


which is true if 

sin 0 = sin d. 

It is customary to omit the connecting phrases and write only the 
equations in this style of proof. 

As one more example we show that 

sec 2 A + esc 2 A = sec 2 A esc 2 A. 

We first express the equation in terms of sines and cosines, 

1 _L_1_ 1 

cos 2 A sin 2 A ~~ cos 2 A sin 2 A 

Clearing of fractions, we have 

sin 2 A + cos 2 A = 1. 

It is a good rule to avoid radicals. When some other 
procedure is not clearly indicated, a reduction to sines and 
cosines is usually effective. 

EXERCISES 

Find the values of the other five functions of 9, by means of 
the formulas of § 59, when a function of 9 and the quadrant in 
which 9 terminates are given as follows: 

1. sin 9 = i, first quadrant. 

2 . sin 9 = - 4 =. > first quadrant. 

V2 

3. cos 9 = f, fourth quadrant. 

4. cos 9 = T \, fourth quadrant. 

5. sec 9 = —- 1 /, second quadrant. 

6. sec 9 = — |, second quadrant. 

7. tan 9 = third quadrant. 

8. tan 0 = A, third quadrant. 

9. cos 9 = .7, first quadrant. 

10. cos 9 = J, first quadrant. 

Express the other five functions of 9 in terms of the following: 

11. cos 9. 12. cot 9. 13. sec 9. 14. esc 9. 





106 


TRIGONOMETRY 


Reduce the following expressions to others containing hut 
one function as indicated. Simplify the results by algebraic 
means where this is possible. 


15. Express sin 2 9 + cos 6 in terms of cos 9 only. 

16. Express cos 6 tan 9 in terms of sin 9 only, 

cos 9 . sin 9 


17. Express 


cos 9 — sin 9 cos 9 + sin 9 


in terms of tan 9 


only. 

18. Express cot a 


1 + cos a 


in terms of sin a only. 


Reduce the following expressions to others containing no 
other functions than sines and cosines, and simplify by alge¬ 
braic means where this is possible: 


19. cos 9 tan 9 + sin 6 cot 9. 


21 . 


tan x 

1 — cot x ' 


cot x 
1 — tana; 


20 L±“. 

1 + cot 2 9 

22 tan A -f sec A — 1 
tan A — sec A -f 1 


Prove the following identities: 


23. 

24. 

25. 

26. 

27. 

28. 

29. 

30. 

31. 


tan 6 + cot 6 = sec 6 esc 9. 

1 — cos 9 _ sin 9 

sin 9 1 + cos 9 

sin 2 x sec 2 x + 1 = sec 2 x. 

cos 2 A . . . 

=-•—r = 1 + sin A. 

1 — sin A 

sec A — cos A = tan A sin A. 


sin x 


+ 


1 + COS X 


1 + COS X 
CSC a cot a = 


= 2 esc x. 


cot a + cos a = 


sin x 
cot a + CSC a 
sin a + tan a 
cot 2 a cos 2 a 


(esc 9 — cot 9 ) 2 = 


cot a — cos a 
1 — cos 9 


1 + cos 9 

















FUNDAMENTAL IDENTITIES 


107 


32. 

33. 

34. 

35. 

36. 

37. 

38. 

39. 


cot 9 cos 9 — esc 0 (1 — 2 sin 2 9) = sin 9. 


/ sec a + CSC aV _ 
\ 1 + tan a ) 


tan ex -j- cot ol 


1 + tan 
1 — sin a 


tan a 
= (sec a — tan a) 2 . 


1 + sin a 
sec 4 y — tan 4 y = 1 + 2 tan 2 y. 
sin y (1 + tan y) + cos y (1 + cot y) = sec y + esc y. 


sin 2 A 


cos' 


cot 2 A ' tan 2 A " ta “ 2 A + cot2A ~ L 
cot 2 A — cos 2 A = cot 2 A cos 2 A. 

(1 — sin C — cos C) 2 = 2 (1 — sin C) (1 


cos C). 


63. Addition formulas. It is easy to show that the sine 
of the sum of two angles, a and 0, is not identically equal to 
sin a + sin /3. Thus if a = 60°, jS = 30°, we have 

sin (a + 0) = sin (60° + 30°) = sin 90° = 1, 

while 

sin a + sin j8 = sin 60° + sin 30° = ^ 


It is not so simple a matter to infer what the correct formulas 
are which express functions of a + /3 in terms of functions of 
a and functions of /3. We shall obtain such addition formulas, 
together with corresponding formulas for functions of a — (3, 
in the following sections. For convenience of reference we 
here list these identities, which should be memorized: 


( 1 ) 

( 2 ) 

(3) 

(4) 

(5) 

( 6 ) 


sin (a + p) 
sin (a — P) 
cos (a + p) 
cos (a — P) 

tan (a + P) 
tan (a — p) 


= sin a cos p + cos a sin p. 

= sin a cos p — cos a sin p. 

= cos a cos p — sin a sin p. 

= cos a cos p + sin a sin p. 

tan a + tan p 
1 — tan a tan p 
tan a — tan p 
1 + tan a tan p 








108 


TRIGONOMETRY 


64. Formulas for sin (a + P) and cos (a + P). We shall 

now prove formulas (1) and (3) of the preceding section for 
all positive acute angles a and (3. In Figure 79 we illus¬ 
trate the case where a + (3 is an angle terminating in the 
first quadrant, and in Figure 80 the case where a + (3 ter¬ 
minates in the second quadrant. The reader should observe 
that the directions for making the construction apply equally 
well to both figures, and that the proof does not distinguish 
one case from the other. 

Figures 79 and 80 are to be constructed as follows. First 
draw coordinate axes OX, OY, and a new set OX i, OYi with 
the same origin 0 and such that angle XOX i = a, angle 



XOYi = 90° + a. Construct the angle XiOP = p. From 
a point P on the terminal side of p drop PQ perpendicular 
to OX i, and draw perpendiculars PM and QN to OX. 
Through Q take axes QX 2 , QY 2 , having the same directions 
as OX and OY respectively. Let R be the intersection of 
QX 2 with MP. 

Figures thus constructed give the coordinates of P in the 
XOY system, in the X 1 OF 1 system, and in the X 2 QY 2 
system. The directed segments OM, MP, are the x and y 
coordinates of P in the XOY system, and from the defini¬ 
tions of the sine and cosine we have 












FUNDAMENTAL IDENTITIES 


109 


(1) sin (a + |8) = 


MP 
OP ’ 


cos (a + ff) 


OM 
" OP ' 


These ratios are to be expressed in terms of sines and cosines 
of a and /3. 

Since NQ and ON, QP and OQ, RP and QR are also coordi¬ 
nates in the systems XOY, XiOY h and X 2 QY 2 , respectively, 


(2) 

NQ 

sm a 0Q > 

ON 

COS a Q q > 

(3) 

• « <2 p 

Sm ^ “ OP ’ 

r-°Q 

cos /3 Qp * 

Moreover, 



(4) sin (90 1 

0 , v RP 

+ a) Qp’ 

cos (90° + oi] 

from which 



(5) 

RP 

COS a = QP } 

QR 

sm a Qp 


QR 

QP’ 


The first equation of (1) may now be written 

, N MP MR+RP NQ+RP _NQ , RP 
(6) sin («+/3) — -Qp — - 0p OP^OP' 

The first term of the right member is expressed in terms of 
sin a and cos 0 if we multiply the first equation of (2) by the 
second of (3). This gives 

* NQ OQ NQ 
sin a cos 0 = -QQ • Qp — Qp> 

and similarly the product of the first equation of (5) and the 
first of (3) gives 

RP 


cos a sin = 


OP 


NO RP 

By substituting these values for p-p and Qp in (6) we obtain 




110 


TRIGONOMETRY 


formula (1) of § 63, 

sin (a + j8) = sin a cos 0 + cos a sin /3. 

In order to treat the second of equations (1) in the same 
way we express OM as ON + NM, which is seen to be cor¬ 
rect when the lengths of the directed segments are given 
their proper positive or negative signs. We thus have 

, , ^ OM ON + NM _ ON QR 

COS (a + P) - 0p 0p 0P + 0 p 

ON OQ QR QP 
~ OQ ' OP ^ QP ' OP 

= cos a cos (3 — sin a sin /?. 


*65. Cases where a and p are not both between 0° and 
90°. It remains to show that the formulas and proofs of 



the preceding section apply without change for all angles 
a and j8, whether positive or negative, no matter in what 
quadrants they terminate. Figures 81 and 82 are drawn 
according to the specifications of § 64. In Figure 81 the 
angle a. is between 90° and 180°, j3 is between 180° and 270°, 













FUNDAMENTAL IDENTITIES 


111 


and a + 0 terminates in the fourth quadrant. In Figure 
82 we illustrate a case where /3 is a negative angle. 

If equations (1), (2), (3), and (4) of §64 are true in all 
cases the rest of the proof will clearly hold good. As to 
equations (1) there is no difficulty. There is also no diffi¬ 
culty regarding equations (2) for Figure 82; but in Figure 
81 the triangle ONQ presents an unfamiliar way of defining 
the functions of a. However, if the reader will refer to § 14 
(p. 15), he will observe that the point whose coordinates 
serve to define the sine and cosine of an angle may be taken 
on either the positive or the negative side of the terminal 
line. Thus in Figure 19 (p. 14) the sine of 6 is defined by 
the ratio of M"P" to OP" as well as by the ratio of MP 
to OP. In Figure 81 the point Q is on the negative side of 
OX i, the terminal line of a, but sin a and cos a are still 
defined by equations (2). 

There is no difficulty with equations (3), since OP is 
always positive. With equations (4) we must again take 
account of cases where the denominator QP is negative. 
This occurs in both Figures 81 and 82, where QP has a nega¬ 
tive length on account of the fact that it is an ordinate in the 
XiOFi system.* The positive direction of the line on which 
QP lies must always be taken as that of OY i, which makes 
an angle of 90° + a with the OX-axis, and therefore with 
the OX 2 -axis. It follows that equations (4) remain correct. 

Thus even in cases where OQ or QP is negative, or both are 
negative, the formulas and proofs of § 64 remain valid. The 
student should convince himself of this fact by drawing 
figures for various types of angles. 

The only cases where our proof is open to objection are 
those where either OQ or QP is zero. When this happens 
/3 is one of the quadrantal angles 0°, 90°, 180°, etc. If we 
substitute each of these values for /3 our formulas will be 

* This becomes clear if the page is turned so that OX i is horizontal and 
OYi extends upward. 


112 TRIGONOMETRY 

found to hold, agreeing with the reduction formulas of 
Chapter IV. 

66. Formulas for sin (a - p) and cos (a - P). We 

easily deduce formulas (2) and (4) of § 63 from (1) and (3), 
which we have shown to hold for all values of a and ft 
Thus, since formula (1) holds whether (3 is positive or neg¬ 
ative, we can substitute —/3 for ft so that we have 

sin (a + (-ft)) = sin a cos (-ft + cos a sin (-ft. 
From § 49 (p. 83) 

cos ( — ft = cos ft sin ( — ft = —sin ft 

and by making these substitutions in the preceding identity 
we obtain the desired formula (2) of § 63 (p. 107). 

sin (a — ft = sin a cos jS — cos a sin ft 
Similarly, from 

cos (a + (-ft) = cos a cos (-ft - sin a sin (-ft, 
we obtain 

cos (a — ft = COS a cos j8 + sin a sin ft 

EXERCISES 

By using the addition formula (1) of § 63 we have 

sin 75° = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30° 

By similar use of formulas (1) to (4) of § 63, hut without using 
the Tables , find the values of the following: 

1. cos 75°. 2. cos 15°. 3. sin 15°. 4. sin (-15°). 

Apply the addition formulas to the following expressions and 
reduce to numerical values, checkihg results: 

5. (a) sin (60° + 30°); (b) cos (45° + 45°); 

(c) cos (60° - 60°). 


FUNDAMENTAL IDENTITIES 


113 


6. (a) sin (90° - 30°); (b) sin (180° + 30°); 

(c) cos (90° + 45°). 

7 . (a) sin (270° - 45°); (b) cos (180° - 30°); 

(c) cos (270° + 60°). 

8. Apply the appropriate addition formula to sin (18O°+0) 
and show that the result agrees with the reduction formula 
for sin (180° + 0). 

Proceed as indicated in Exercise 8 with the following: 

9 . cos (90° + 0). 10 . sin (180° — 0). 

11. cos (180° + 0). 12. sin (270° — 0). 

13 . Given that a and are positive acute angles for which 
cos a = | and sin ]8 = find sin (a + /3) and cos (a — j8). 

14 . Given that a and 0 are positive acute angles for 

which sin a — and cos ,8 = £, find the values of cos (a + (3) 

and sin (a: — /3). 

15 . By use of the Tables find the approximate numerical 
difference between 

(a) sin (47° - 32°) and sin 47° - sin 32°, 

(b) cos (47° + 32°) and cos 47° + cos 32°. 

Prove the identities: 


16 . sin (45° — 0) 


cos 0 — sin 0 

V2 


17 . cos (60° + 0) 

18 . sin (30° + 0) 


cos 0 — a/ 3 sin 0 
2 

cos 0 + %/3 sin 0 
2 


19 . cos (45° + 0) 


cos 0 — sin 0 

V2 


20. sin ( A + B) cos B — cos ( A + B) sin B = sin A. 

21. cos ( A — B) cos B — sin (A — B) sin B = cos A. 

22. sin (x + y + z) = sin x cos y cos z + cos x sin y cos z 

+ cos x cos y sin z — sin x sin y sin 5!. 






114 


TRIGONOMETRY 


23 . cos (x + y + z) = cos x cos y cos z’— sin x sin y cos 2 


— sin x cos y sin z — cos x sin y sin z. 


24 . Prove the formulas for sin ( a + 0) and cos (a + /3), 
drawing the figure, when a and (3 are each angles between 
90° and 180°, and a + 0 is less than 270°. 

25 . Prove the formulas for sin (a — /?) and cos (a — fi), 
drawing the figure, when a is between 90° and 135°, and 0 is 
between 45° and 90°. 

26 . If x, y are the coordinates of P in the XOY system, 
and xi, yi its coordinates in the X 1 OY 1 system as described 
in § 64, prove that 

x = Xi cos a — yi sin a, y = x\ sin a + yi cos a. 

67. Formulas for tan (a + P) and tan (a — p). From 

formula (2) of § 59, and formulas (1) and (3) of § 63, we have 


sin (a + (3) _ sin a cos (3 + cos a sin /3 
cos (a + 0) cos a cos (3 — sin a sin 0 


tan (a + j8) = 


We can express the last fraction in terms of tan a and tan 0 
if we divide both numerator and denominator by cos a cos /?. 
We thus obtain 



tan (a + 0) = 


sin a sin 0 
cos a cos j 3 


sin a sin 0 
cos a ' cos ft 


sin a sin /3 
cos a cos /3 


From this identity we at once derive formula (5) of § 63 


(p. 107), 


(1) tan (a + /3) = 


tan a + tan (3 


1 — tan a tan 0 











FUNDAMENTAL IDENTITIES 


115 


If we treat in the same way the identity 

tan (a — p) = S * n ~ ft) _ sin a cos (3 — cos a sin ft 
cos (a — p) cos a cos p + sin a sin P ’ 


we obtain formula (6) of § 63, 


tan (a — p) 


tan a — tan p 
1 + tan a tan p 


EXERCISES 

By expressing the given angles as sums or differences of 45° 
and 30° and using formulas (5) and (6) of § 63, but without 
using the Tables, find the values of the following: 

1. tan 75°. 2. tan 15°. 

3 . Apply the addition formulas to the following expres¬ 
sions and reduce to numerical values, checking results: 

(a) tan (60° + 60°); ( b ) tan (60° - 60°); 

(c) tan (180° - 30°). 

4 . By means of the formulas of the preceding section 
obtain the reduction formulas for 

(a) tan (180° + 9); (b) tan (180° - 9); 

(c) tan (360° - 9 ). 

5 . If tan x = f, cos y = rb an d x and y are positive 
acute angles, find the values of tan (x + y) and tan (x — y). 

6. If sin x = |, tan y = 1 i, and x and y are positive acute 
angles, find the values of tan (x + y) and tan (x — y). 

7. Show by comparing values taken from the Tables that 
tan 40° + tan 20° is not equal to tan (40° + 20°). 

8. Show by comparing values taken from the Tables that 
tan 70° — tan 30° is not equal to tan (70° — 30°). 

Prove the identities: 

9 . tan (45° + B) = J + 






116 


TRIGONOMETRY 


10 . 

11 . 

12 . 

13 . 

14 . 

15 . 

16 . 


tan (45° — 0) 
tan (30° A.') = 


1 — tan 0 
1 + tan 0 

1 -f- V3 "tan A 
V3 — tan A 
tan A — V3 
1 + V3 tan A 
tan y 


— tan x. 


tan (A — 60°) = 

tan (x + V) _ 

1 + tan (x + y) tan y 
tan (x - y) + tan y 
1 — tan (x — y) tan y 

, t cot a cot 0 - 1 

COt (a + j8) 


cot (a — (3) = 


cot a + cot j8 
cot a cot j3 + 1 
cot a — cot 


68. Formulas for the double angle. When jS is taken 
equal to a in the formulas for sin (a + j8), cos (a + /3), 
tan (a + j8), we obtain identities which express functions of 
2 a in terms of functions of a. 

For example, 

COS (a + a) = cos a cos a — sin a sin a 
is equivalent to 

cos 2 a = cos 2 a. — sin 2 a. 


The double angle formulas thus obtained are (1), (2a), and 
(3) of the following set. Formula (2b) is derived from (2a) 
by the substitution sin 2 a — 1 — cos 2 a ; formula (2c) by the 
substitution cos 2 a = l — sin 2 a. 


(1) 

sin 2 a = 2 sin a cos a. 

(2a) 

cos 2 a = cos 2 a — sin 2 

(2b) 

cos 2 a = 2 cos 2 a — 1. 

(2c) 

cos 2 a = 1 — 2 sin 2 a. 


2 tan a 

(3) 

tan 2 a = z -t— s— • 

1 — tan 2 a 










FUNDAMENTAL IDENTITIES 


117 


Example. — Find the sine, cosine, and tangent of 120° 
by means of the double angle formulas. 

sin (120°) = sin (2 X 60°) = 2 sin 60° cos 60° 


a/3 1 _ V3 
2 2 ~ 2 




tan (120°) = 


69. Formulas for the half-angle. Since the angle 2 a in 
the preceding formulas is any angle whatever of which a 
is half, the formulas are equally true when a is replaced con¬ 
sistently by any other symbol denoting an angle. If, for 
example, we replace a by 2 a in identity (1) of the preceding 
section, we have 


sin 4 a = 2 sin 2 a cos 2 a. 


Results of especial interest are obtained by replacing a by 
a/2 in formulas (1), (2b), (2c) of § 68. These formulas then 


become 


~ . a a 
sin a = 2 sin ^ cos 2 ’ 


( 1 ) 



( 2 ) 


(3) 


If we solve (3) for sin (a/2) and (2) for cos (a/2) we obtain 
the first two of the following formulas for the half-angle , the 
third being obtained by dividing the expression for sin (a/2) 
by the expression for cos (a/2): 



(4) 


2 






118 


TRIGONOMETRY 


a , /l + cos a 
(5) coSg = ±y-2- 

a , /1 — cos a 
( 6a ) tan 2 “ ^ V 1 + cos a * 

Whether the positive or negative sign is to be taken in each 
case depends on the quadrant in which the angle a/2 ter¬ 
minates. For example, if a = 420° the angle a/2 terminates 
in the third quadrant; its sine and cosine are negative and 
its tangent is positive. Formula (4) gives 

. 4 /]—cos420° 4 /l-cos60° 1 

sm210 =-y- 2 - =_ V-2- =— V 2 _ 2’ 


and the other formulas would similarly give numerical values 
for cos 210° and tan 210°. 

The following are better formulas, for some purposes, 
than (6a): 


(6b) 



1 — cos a 
sin a 


(6c) 


a sin a 

tan ^—;-. 

2 1 + cos a 


The former of these identities is easily verified if we sub¬ 
stitute in the expression on its right the values of sin a and 
cos a given by (1) and (3). We thus have 


1 — cos a 
sin a 



^ . a a .a a a 

2 sin ^ cos 2 2sm - cos ^ cos ^ 



To prove (6c), substitute in its right member the values of 
sin a and cos a given by formulas (1) and (2) (see also Exer¬ 
cise 24, p. 106). 

Examples. — 1. Find the sine, cosine, and tangent of 
a/2 if a is an angle between 360° and 450° for which tan a = 2. 

















FUNDAMENTAL IDENTITIES 


119 


Here a terminates in the first quadrant, and a/2 in the third. 
This determines the sign of cos a and of the functions of a/2. We 
have 

1 1 ,1 

COS a = = - = -f- — j 

sec O' + Vl + tan 2 o- V5 

/- 2 

sin a = vl — cos 2 a = “7= » 
v5 

.a , /l — COS a 4 / ^5 4/5-V5 

sin 2 = “V-2 - = "V “ 2 “ 

a ./l+COSa i/5 + V5 

cos 2 = V-2- = ’ 

1 “ "7= 

a 1 — COS a V5 V5 — 1 

tan 2 = sin a _ _2_ " 2 

V5 

2. Prove the identity ! , C ° S W = tan 2 A. 

J 1 + cos 2 A 


Of the many possible proofs we shall give two. First, we observe 
that the angle in the left member is 2 A, in the right A. Hence we 
use formulas to express each in terms of the same angle. Formula 
(2a), (2b) or (2c) will serve to change the angle from 2 A to A in 
the left member. If we use (2c) in the numerator and (2b) in the 
denominator the first terms cancel; we have 

1 - cos 2 A 1 - ( 1-2 sin 2 A) 

1 + cos 2 A “ 1 + (2 cos 2 A - 1) 

_ 2 sin 2 A 
~ 2 cos 2 A 
= tan 2 A. 


A second method is suggested if we observe that the identity re¬ 
sembles formula (6a). Let us substitute A = a/2; we are to prove 


that 


1 — COS a 
1 + COS a 


tan 2 ^ 























120 


TRIGONOMETRY 


This follows at once from (6a), by interchanging members in that 
formula and squaring. 

In proving identities it is usually desirable to express all 
angles in terms of one angle, and all functions in terms of 
one or two functions. It is best to avoid radicals when 
possible. 


EXERCISES 

1. Find the values of sin 2 a, cos 2 a, tan 2 a, sin a/2, 
cos a/2, tan a/2, without using the Tables, from the follow¬ 
ing data: 

(a) a is between 0° and 90°, and sin a = f. 

(b) a is between 450° and 540°, and tan a = — 

2. Proceed as in Exercise 1, with the following data: 

(a) a is between 0° and 90°, and cos a = 

(b) a is between 540° and 630°, and cot a = ^ 2 -. 

3. Substitute a = 30° in the double angle formulas and 
thus obtain the numerical values of sin 60°, cos 60°, tan 60°. 

4. Substitute a = 45° in the formulas for sin 2 a and 
cos 2 a , and thus obtain the numerical values of sin 90° and 
cos 90°. 

5. Substitute a = 30° in the formulas for the half-angle 
and thus obtain the numerical values of the functions of 
15°. 

6. Substitute a = 45° in the formulas for the half-angle 
and thus find the numerical values of the functions of 
22 |°. 


Prove the identities: 


7 . 

8 . 


(sin 0 + cos 0) 2 = 1 + sin 2 0. 
2 tan A 


sin 2 A = 


1 + tan 2 A 



FUNDAMENTAL IDENTITIES 


121 


9. sec a — ■ 


sec 2 
2 — sec 2 ^ 


10 . 


1 + tan x 1 + sin 2 x 


1 — tan x cos 2 x 
11 . 1 + tan A tan 2 A = sec 2 A. 


12. tan 


( 45 ”+i)- 


sec 0 + tan 0. 

13 . cos 0 + sin 2 0 cot 0 = 1 + cos 0 + cos 2 0. 

14 . 2 cot 0 = ^cot | — tan Q • 

15 . cos 3 x + sin 3 x = (1 — J sin 2 x) (cos £ + sin x). 
2 


16. 1 + cos 2 x = 


(1 + tan x tan ? 


a — (3 sin a — sin f 


17 . tan ^ cos a _j_ cos £ 

18. sin 3 a = sin {2 a A a) = 3 sin a — 4 sin 3 a. 

19. COS 3 a: = COS {2 a A a) =4 cos 3 a — 3 cos a. 

. . 3 tan a — tan 3 a 

20. tan 3 a = tan (2 qj + a) — _ 3 ^ an2 ^ 

. . , . A A 0 . ,A A 

21. sm 2 A =4 sin cos — 8 sin 3 cos —. 

A A 

22. cos 2 A = 1 — 8 sin 2 - 2+8 sin 4 — • 

23. Prove that the area of a right triangle with right angle 
at C is i c 2 sin 2 A. 

24. For the right triangle of Exercise 23, prove that 

a 


, A 

tan-g 


b A c 


70. Products which are equal to sums or differences of 
two sines or two cosines. From the addition formulas of 
§ 63 we obtain, by addition and subtraction, 










122 


TRIGONOMETRY 


(1) sin (a + p) + sin (a - P) = 2 sin a cos p. 

(2) sin (a + P) - sin (a - P) = 2 cos a sin p, 

(3) cos (a + p) + cos (a - p) = 2 cos a cos p, 

(4) cos (a + p) - cos (a - p) = -2 sin a sin p, 


If we read these formulas from right to left they express 
products of a sine or cosine of one angle by the sine or cosine 
of another as equal to one-half of sums or differences of 
sines or cosines. 

For purposes of computation it is often more convenient 
to deal with products of functions than with their sums. 
The four formulas express sums as products, but a change of 
notation is advantageous. Let us make the substitutions 

A = a + ft B = a- (3. 


By adding and subtracting these equations we obtain 
2 a = A A B, 2 (3 = A B’ 

A+B n A-B 

a = jr— > P — —o- 


When a and 0 are replaced by these values, formulas (1) to 
(4) become 

* . -»-v n . A + B A-B 

( 5 ) sin A + sin B =2 sin — ^— cos — 2 — * 

.„ 0 A+B.A-B 

(6) sinA - smB = 2 cos—^— sm — 2 — ’ 

* t~> n A + B A-B 

(7) cos A + cos B = 2 cos —^— cos — 2 — ’ 

__ 0 . A + B. A-B 

(8) cos A — cos B = —2 sm— ^— sin —o— * 


A good way to memorize these identities is to put them in 
words. Thus formula (5) is equivalent to the statement: 
The sum of the sines of two angles is equal to twice the sine 












FUNDAMENTAL IDENTITIES 


123 


of half the sum of the angles, multiplied by the cosine of half the 
difference. 

Examples. — 1. Prove that sin 40° + sin 20° = sin 80°. 
By formula (5), 


sin 40° + sin 20° = 2 sin 


40 c 


+ 20° 40° - 20 c 

—-cos-- 


= 2 sin 30° cos 10° 

= 2 • ~ • cos 10° 

= cos 10° = sin 80°. 


2 . 


Prove that 


sin A — sin B 
cos A — cos B 


— cot 


A + B 
2 


By formulas (6) and (7), 


sin A — sin B 
cos A — cos B 


A +B . A -B 
2 cos — t ,— sin —2— 

. A + B . A - B 

-2 sin —2— sin —2— 

A +B 

-COS-- 


. A +B 
sm 2 

A B 
— cot—2- 


EXERCISES 

1. Prove the following relations without using the 
Tables, then check by referring to the Tables: 

(a) sin 30° + sin 60° = V2 cos 15°. 

(b) cos40° - cos 20° = -cos 80°. 

(c) sin 75° - sin 15° = cos 45°. 

(d) cos 75° + cos 45° = cos 15°. 
















124 


TRIGONOMETRY 


2. Express 2 sin 3 0 cos 0 as the sum of two sines. 

3. Express 2 cos 8 0 sin 0 as the difference of two sines. 

4. Express sin 5 A sin 2 A as half the difference of two 
cosines. 

5. Express cos 2 A cos 3 A as half the sum of two cosines. 


Prove the following identities: 

6. sin 3 0 — sin 0 = 2 cos 2 0 sin 0. 
cos 7 0 + cos 50 = 2 cos 6 0 cos 0. 


7 . 

8 . 

9 . 

10 . 

11 . 

12 . 


sin A + sin B , A + B 
= tan 


cos A + cos B 
sin A + sin B 


sin A — sin B 
cos 2 a — cos a 


= tan 


= tan 


2 

A + B 
2 
3 a 

T' 


cot 


A — B 


sin a — sin 2 a 
cos a (cos a — COS 3 a) = sin a (sin a + sin 3 a). 


sin A + cos B 
Hint. Express cos B as sin (90° 


2 sin ^45°+ 


A — B 


^ cos ^ 


A+ 5 


B ). 



13 . 

14 . 

15 . 

16 . 

17 . 

18 . 
19 . 


20 . 


21 . 


22 . 

23. 


sin A - cos B = - 2 cos ^45°+sin ^45° - * 

sin x + sin 2 x + sin 3 x = sin 2 £ (1 + 2 cos x). 
cos (45° + a) + cos (45° — a) = \/2 cos a. 
sin (60° + a) — sin (30° — a) = \/2 sin (15° + a:), 
cos (a — 13) _ 1 + tan a tan ft 
cos (a + j8) 1 — tan a tan /3 

cot %. — 2 cos 2 %. cot x — sin x. 

2 2 

cot 2 0 — esc 2 0. 


cos 


( 0 0 V 

sin - + cos ^ J 


sin A — sin 2 A + sin 3 A =4 sin \ A cos A cos f A. 


tan \ A = 


1 + sin A — cos A 


1 + sin A + cos A 
8 sin 3 a cos a = 2 sin 2 a: — sin 4 a. 















FUNDAMENTAL IDENTITIES 


125 


Prove that if A, B, C are angles of a triangle, the following 
identities hold: 

24 . sin A + sin B + sin C = 4 cos \ A cos J B cos \ C. 

Hint. C = 180° - (A + B). 

25 . sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C. 

26 . tan A + tan B + tan C = tan A • tan B • tan C. 

27 . In a triangle ABC the line AD is drawn perpendicular 
to BC, and D falls between B and C. If angle DAB = a, 
angle CAD — (3, AD = h, prove that 

BC = + 

cos a COS /3 

Does this formula hold if D does not fall between B and C? 

28 . An observer sees from a point A that the angle of 
elevation of the top, B , of a flagpole BC is a. He travels 
backward in the plane ABC, and from a point D on the same 
horizontal level as A observes that the angle of elevation of 
B is j 3. If AD = a, prove that the height h of the top of the 
flagpole above the horizontal level of A is given by the for¬ 
mula 




CHAPTER VI 

RADIAN MEASURE. INVERSE FUNCTIONS 

71. The radian. So far we have used only degrees, 
minutes, and seconds in measuring angles. Another unit, 
the radian, is more convenient in certain problems which will 
be considered in the following sec¬ 
tions. 

In defining sin 9 and cos 6 we noted 
(§ 14, p. 16) that the ratio of MP to 
OP would be the same no matter how 
long OP was taken; and similarly for 
the ratio of OM to OP. In Figure 83, for example, we have, 
by similar triangles, 

MiPi M 2 P 2 0M 1 0M 2 

OP i 0P 2 ’ OP i 0P 2 ’ 

According to a proposition of plane geometry, it is also true 
that 

arc AiPi _ arc A 2 P 2 
OPi ” 0P 2 9 

where the two circular arcs have their centers at 0. In 
other words, for an angle whose vertex is at the center of a 
circle the ratio of subtended arc to radius is the same no 
matter how long we make the radius; it can be considered an 
additional function of the angle. 

Though sine and cosine determine an angle, they do not 
serve to measure it as degrees, minutes, and seconds do; 
when we double an angle we double its degree measure, but 
we do not, in general, double its sine or cosine. However, 
the ratio of arc to radius does have the property of being 

126 



Fig. 83 








RADIAN MEASURE. INVERSE FUNCTIONS 127 


directly proportional to the angle. This follows from the 
proposition of plane geometry which states that angles whose 
vertices are at the center of a given circle are proportional to 
the intercepted arcs. It follows that if we double 9 in 
Figure 83 we shall double arc AiPi, and we shall therefore 
have doubled the ratio of arc AiPi to 
the radius OP i. 

The ratio of arc AiPi to the radius 
OP i is called the radian measure of 
the angle AiOPi. c 

The angle whose radian measure is 1 
is called the radian. The arc which 
subtends it has a length equal to that 
of a radius. The radian measure of 
an angle is the number of radians it contains; for, in Fig¬ 
ure 84, 

, 8 arc BC zBOC 

(1) radian measure of 9 = - = ^rjp - z AOP 

= number of times the radian is 
contained in angle BOC. 

72. Relations between radians and degrees. By means 
of equation (1) at the end of the preceding paragraph we can 
compare measurements in radians and degrees. Thus if 9 
subtends a semicircumference we have 9 — 180°. On the 
other hand, since s = xr, we have from equation (1), 

9 = — = x radians. 
r 

By comparing these two values of 9 we obtain the relation 

(1 ) 180° = tt radians. 

It follows that 

(2) 1° = radians = .017 4533 radians, 






128 


TRIGONOMETRY 


and 

180° 

(3) 1 radian = — = 57.29578° 

7r 

= 57° 17' 45", 

these results being correct to the number of figures given. 

Hereafter, if the measure of an angle is given as n we will 
understand that this means n radians, unless the contrary 
is clearly indicated.* 

In the second column of each page of Table II will be 
found the radian equivalent of the degrees and minutes in 
the first column. By the use of this Table, with interpola¬ 
tion, we can convert the measure of an angle from degrees 
and minutes into radians with four-place accuracy; and 
vice versa we can change four-place radian measure into 
degrees and minutes. It will be useful, however, to con¬ 
sider examples in which formulas (2) and (3) are used directly. 

Examples. — 1. To express 5 radians in degrees and 
minutes. 

We have from (3), 

5 radians = 5 X (57° 17' 45") 

= 285° + 85' + 225" 

= 285° + (1° 25') + 4' (approximately) 

= 286° 29'. 

2. Express 7 t/ 6 radians in degrees. 

From (2) we have 

TV TV 180° 

^ radians = ,, X - = 30 . 

O O 7T 

3. Express 20° 23' in radians. 

We shall give the results in two forms, the first in terms of x, the 
other a decimal. 

* Some authors use the notation n r for n radians, but this is apt to be 
confused with the symbol for n to the rth power. 


RADIAN MEASURE. INVERSE FUNCTIONS 129 


(a) 20 




1223 7T 
60 X 180 
1223 


(b) 


20° 23' = 20 X .0174533 + ^ X .0174533 

= .34907 + .00669 
= .3558 radians (to four places). 


Result (b) could have been obtained from Table II by interpo¬ 
lating between the given values 


20° 20' = .3549 radians, 
20° 30' = .3578 radians. 


For 20° 23' the correction which should be added to .3549 would be 
3/10 X 29 =9, giving .3558 as in (b). 

73. Length of circular arc. The equation at the end of 
§ 71 can be written in the form 


( 1 ) 


Note that if 9 is not given in radian measure, it must be 
so expressed before this formula is used. When any two of 
the three quantities in (1) are given, the third is obtained by 
solving (1). 

Examples. — 1. If the radius of a circle is 18‘ft., find in 
terms of ?r the arc subtending an angle of 15°. 

We first reduce 15° to radians. 


15° = 15 XjJq = ^radians. 


Formula (1) then gives 


3 



i30 


TRIGONOMETRY 


2. Find the number of degrees and minutes in an angle 
whose vertex is at the center of a circle if the radius is 2.0000 
and the subtending arc is 2.3566. 

If we solve (1) for the radian measure 8 of the required angle, 
we have 


s 2.3566 
“ r ~ 2.0000 


1.1783 radians. 


By means of Table II we reduce this radian measurement to degrees 
and minutes, and obtain the result 


8 = 67° 31'. 


EXERCISES 

1. Express the following in radian measure, giving results 
in terms of ir: (a) 30°; (b) 45°; (c) 180°; (d) 25 15'; 
(e) 73° 27'; (f) 169°. 

2. Proceed as in Exercise 1 with the following: (a) 60°; 

(b) 90°; (c) 270°; (d) 37° 45'; (e) 84° 18'; (f) 137°. 

3. Reduce the degree measures of each part of Exercise 1 
to radian measure in decimal form without using the Tables. 

4. Proceed with each part of Exercise 2 according to the 
directions in Exercise 3. 

6. The following are radian measures; reduce them to 
degrees and minutes without using the Tables: (a) ^ ; (b) ^ ; 

(c) (d) (e) 2.5; (f) .6250. 

6. Proceed as in Exercise 5 with the following radian 
measures: (a) g ; (b) tt; (c) ^ ; (d) —g—; (e) 3.2; (f) .5241. 

7. Express the following in radian measure, using the 
Tables and giving results to four decimal places: (a) 25° 17'; 
(b) 73° 42'; (c) 143° 24'. 

8. Proceed as in Exercise 7 with the following: (a) 16° 29'; 
(b) 65° 22'; (c) 169° 17'. 



RADIAN MEASURE. INVERSE FUNCTIONS 131 


9 . Express the following radian measures in degrees and 
minutes, using the Tables: (a) .1200; (b) 1.3027; (c) 2.4050. 

10. Proceed as in Exercise 9 with the following radian 
measures: (a) .3030; (b) 1.2452; (c) 3.1080. 

11 . An angle at the center of a circle of 2 ft. radius inter¬ 
cepts an arc of 3 ft. Find the measure of the angle, first in 
radians, then in degrees and minutes, assuming the measure¬ 
ment of radius and arc to be exact. 

12. Proceed as in Exercise 11 if the radius is 10 ft. and the 
intercepted arc is 23 ft. 

13. The radius of a circle is 1.500 ft. Find the arc which 
subtends ai$ angle of 65° O'. 

14. The radius of a circle is 1.250 ft. Find the arc which 
subtends an angle of 237° 12'. 

15. An angle of 2.500 radians at the center of a circle in¬ 
tercepts an arc just 15 in. long. Find the radius. 

16. An angle of 217° 0' at the center of a circle intercepts 
an arc of length 235.0 yd. Find the radius. 

17. If the earth’s radius is 3960 mi., how far is it on the 
earth’s surface from a point in latitude 41° 10' to the nearest 
point on the equator? 

18. Show that if 6 is the radian measure of a positive acute 
angle, then sin Q < d. Is this true when 6 is greater than 

tt/2? 

19. Show that if 6 is the radian measure of a positive acute 
angle, then tan 6 > 6. 

Hint. If two points A, B, lie on a circle, and the tangents at A 
and B intersect at C, then AC + CB is greater than arc AB, provided 
the latter is less than the semicircumference. 

20. Two points A and B are on the equator of a globe, 
and their longitudes are 19° 50' E and 43° 10' E respectively. 
The arc AB is found to be 3.250 in. Find the radius of the 
globe. 

21. A belt passes tightly, without crossing, over two wheels 
which are in line with centers 22 ft. 7.5 in. apart. The 


132 


TRIGONOMETRY 


diameter of the larger wheel is 6 ft. 6.0 in., that of the smaller 
is 2 ft. 4.2 in. Find the length (a) of the part of the belt in 
contact with the larger wheel; (b) of the part in contact 
with the smaller wheel; (c) of the whole belt. 

22. Give the lengths asked for in Exercise 21, for a belt 
that is crossed. 

*74. Areas of segment and sector of a circle. A radius 
of a circle revolving from an initial position OA sweeps out a 
sector whose area is directly propor¬ 
tional to the angle A OB through which 
the radius has turned. Hence, if we 
compare the area of the sector OACB, 
A whose central angle is 9 radians, with the 
area of the semicircle, whose central 
angle is ir radians, we have 

area of sector OACB _ 9 
area of semicircle ir 

If the radius is of length r, the area of the semicircle is 
7rr 2 /2. The preceding equation, when solved for the area 
of OACB , has on its right side 

- X area of semicircle = - • —• = \ r 2 6 . 

7r 7 T Z 

Hence we have the formula 

(1) area of sector OACB = J r 2 6. 

The area of segment ACB (shaded in Figure 85) is given 
by the relation 

area of segment ACB = area of sector OACB — area of tri¬ 
angle OAB. 

If OA is taken as the base of triangle OAB, it is easy to see 
that the altitude is OB sin 9, and hence, 

area of OAB = \ OA • OB • sin 9 = \ r 2 sin 9. 


B 





RADIAN MEASURE. INVERSE FUNCTIONS 133 


Thus the right side of the expression for the area of the 
segment becomes 

i r 2 0 - i r 2 sin Q = £ r 2 (0 - sin 0), 
and we have the formula 

(2) area of segment ACB = J r 2 (0 — sin 0). 

In using both formulas (1) and (2), it is important to 
remember that 0 is the radian measure of the angle. 

*75. Velocity of a point moving in a circle. A point P 
is said to move on the circumference of a circle with uniform 
linear velocity of magnitude v = s/t if it traverses an arc s 
in time t and the ratio of s to t is constant. The angular 
velocity of P is 6ft, where 0 is the angle generated by the 
radius OP when P traverses the arc s. By the angular 
velocity of OP we mean the same thing as the angular velocity 
of the point P. It is customary to designate angular ve¬ 
locity by the Greek letter co (omega). We may measure co 
in units either of degrees, radians, or revolutions per minute 
or second. 

When 0 and co are given in terms of radians, equation (1) 
of § 73 yields a formula connecting v with co; for if both sides 
of that equation are divided by t, we have 

s 0 

i = r t’ 

hence 

v = rco. 

Example. — A flywheel 10 ft. in diameter makes 100 revo¬ 
lutions per minute. For a point P on its rim find the linear 
velocity in feet per minute and the angular velocity in 
radians per minute. 

The circumference of the wheel is 10 x ft., hence 
s 100 X 10 7r 

= 7 = 1 


v 


= 1000 x ft. per min.; 



134 


TRIGONOMETRY 


also, since a radius generates an angle of 2 x radians for each revolu¬ 
tion, 

co = 100 revolutions per min. = 200 n r radians per min. 

These results check with the relation v = r«. 

EXERCISES 

1 . Find the area of a sector whose angle is 18°, if the sub¬ 
tending arc is 12 ft. long. 

2 . Find the area of a sector whose angle is 125°, if the sub¬ 
tending arc is 25 ft. long. 

3. Find the area of a segment whose bounding arc is 
16 in. long, in a circle whose radius is 1 ft. 

4 . Find the area of a segment if the chord that forms part 
of its boundary is 26 in. long, and is 11 in. from the center of 
the circle. 

5. A horizontal cylindrical tank, 15 ft. long and 4 ft. in 
diameter, is partly filled with water so that the greatest 
depth is 15 in. How many gallons of water are there in the 
tank if the volume of a gallon is 231 cu. in.? 

6 . Find v in inches per minute and co in radians per second 
for a point at the end of the minute hand of a clock if the 
hand is 22.5 in. long. 

7. Solve the problem of Exercise 6 if the hand is 18.6 in. 
long. 

8 . A wheel 9.2 ft. in diameter revolves with uniform angu¬ 
lar velocity of 3.0 radians per second. Find v in feet per 
minute for a point on the rim. 

9. The wheels connected by a belt as described in Exercise 
21 of page 131 rotate uniformly so that the angular velocity 
for the larger wheel is (to three significant figures) 200 
revolutions per minute. What is the angular velocity of the 
smaller wheel in radians per second? 

10. Prove that formula ( 2 ) of § 74 is true when 0 is greater 
than 7 r. 


RADIAN MEASURE. INVERSE FUNCTIONS 135 


76. Inverse trigonometric functions. Principal values. 

Another way of stating that sin 9 is equal to a is to say that 
9 is an angle whose sine is a, or, more briefly, that 9 is the 
inverse sine of a. This is written 9 = sin -1 a. 

The two equations 

sin 0 = a, 9 = sin -1 a, 
mean exactly the same thing. 

We define similarly the other inverse functions cos -1 a, 
tan -1 a, cot -1 a, sec -1 a, esc -1 a. Another notation for these 
functions is arc sin a, arc cos a, etc. 

The student should be on his guard against interpreting 
sin -1 a as the — 1 power of sin a. Although we write sin 2 a 
for (sin a) 2 , and similarly for other powers, the — 1 power 
should always be written as (sin a) -1 ; sin -1 a always means 
the inverse sine of a. 

The inverse functions are many-valued . For example, 
since 

sin 30° = sin 150° = sin (360° + 30°) = • • • = 
we have 

sin -1 1 = 30°, 150°, 360° + 30°, . . . 

The problem of finding the values of sin -1 a is the same as 
that of solving for 9 the equation sin 9 — a. If a is between 
0 and 1, we have seen in § 55 (p. 92) that there is one and 
only one solution, 9 = 9i, between 0° and 90°, and one and 
only one, 9 = 180° — 9± } between 90 and 180 j all others 
are obtained from these two by adding or subtracting mul¬ 
tiples of 360°. Among the infinitely many values of sin- 1 a, 
we distinguish as the principal value that one which lies be¬ 
tween 0° and 90° when a is between 0 and 1. A convenient 
way to designate this principal value is to write it Sin -1 a 
(with the initial S capitalized). 


136 


TRIGONOMETRY 


When we consider negative as well as positive values of a, 
we find that there is one and only one value of sin -1 a be¬ 
tween — 90° and +90° for each value of a between —1 and 
+ 1 .* We call this the principal value and designate it by 
the notation Sin -1 a; its range is from —90° to +90°. 

To define principal values for the other inverse functions, 
we specify for each a range of angles in which the principal 
value must lie. If a is any number for which a given inverse 
function has a meaning, then the range for that function 
should be such that one and only one value of the function 
in that range corresponds to each value of a. We have seen 
that this is true of the range from —90° to +90° for Sin -1 a. 
This range would not serve for Cos -1 a, since if a is a negative 
proper fraction the equation 9 = cos -1 a, or its equivalent, 
cos 9 = a, is satisfied only by angles terminating in the sec¬ 
ond or third quadrant. A range from 0° to 180° would, 
however, be appropriate, and this we adopt. The range for 
Csc -1 a is taken the same as for Sin -1 a, and for Sec -1 a the 
same as for Cos" 1 a. For Tan" 1 a we take the same range 
as for Sin -1 a. If we take for Cot -1 a the same range as for 
Cos -1 a, we complete the scheme of the following table in 
which it will be noted that the range of principal values for 
three inverse functions is from —90° to +90°, while for the 
three corresponding cofunctions the range is from 0° to 180°. 
To the right we indicate the values of a for which each in¬ 
verse function has a meaning. 


— 90° ^ Sin -1 a ^ 90°, 
0° ^ Cos- 1 a ^ 180°, 
-90° < Tan" 1 a < 90°, 
0° < Cot- 1 a < 180°, 
0° ^ Sec- 1 a ^ 180°, 
-90° ^ Csc- 1 a ^ 90°, 


-1 ^ a ^ 1; 

1 ^ a ^ - 1 ; 
a may have any value; 
a may have any value; 
a ^ 1 ora ^ — 1; 
a ^ -1 ora ^ 1. 


* The symbol sin -1 a has no meaning for us unless a is between —1 and 
+1 since a is. by definition, the sine of an angle. 


RADIAN MEASURE. INVERSE FUNCTIONS 137 


77. Determination of all values of an inverse trigonometric 

function. If a is positive the Tables give us the principal 
value of each inverse function of a. For example, we would 
find Sin -1 .5640 by looking on page 7 of Table II, where it 
is given that an angle whose sine is .5640 is 34° 20' = .5992 
radians. 

To find the principal value of an inverse function of a 
negative number —a we may proceed as follows: 

Find the principal value 0i of the inverse function of +a, using 
the Tables if necessary; then — 0i is the principal value of the 
inverse function of —a if the function is the inverse sine, 
tangent, or cosecant; otherwise 180 ° — 0i is the value to be 
used (or tt — 0i, in radians). 

This rule follows from the reduction formulas of Chapter 
IV and from the definitions of principal values. For example, 
from the reduction formulas if sin 0i = a then sin (— 0i) 
= —a; or if cot 0i = a then cot (180° — 0i) = — a. Hence 
—0i = sin -1 ( — a), and 180° — 0i = cot -1 ( — a). Finally, 
these are both principal values since, on account of the fact 
that 0i is a positive acute angle, they are in the ranges given 
by formulas (1) of § 76. 

Having thus found the principal value 0 of an inverse 
function of a, we observe that a secondary value of that in¬ 
verse function will be: 

180° — 0 for sin -1 a and esc -1 a; 

— 0 for cos -1 a and sec -1 a; 

180° + 0 for tan -1 a and cot -1 a. 

We can prove that these are values of the inverse functions 
indicated by using the reduction formulas. For example, 
since cos (— 0) = cos 0, it follows that if cos 6 = a, then 
cos (-0) = a, and both 0 and -0 are values of cos" 1 a. 

When the principal value and the secondary value of an 
inverse function have been found, all other values of that 


138 


TRIGONOMETRY 


inverse function are obtained by adding or subtracting 
multiples of 360° (or 2 7r radians). 

Examples. — 1. Find all values of tan -1 ( — 2.000), giving 
results in radians. 


We first find from the Tables (with interpolation) that 

88 

Tan- 1 2.000 = 1.1054 + ^ X .0029 = 1.1072. 
145 


Hence 


Tan- 1 (-2.000) = -1.1072, 


and the secondary value of tan -1 (—2.000) is —1.1072 + t. The 
general solution is 


tan- 1 (-2.000) 


-1.1072 
-1.1072 + 7r 


± 2 utt, (n = 0, 1, 2, . . .)• 


2. Find all values of cos -1 (.5000) in degrees. 
Since 

Cos- 1 (.5000) = 60% 
we have 

cos- 1 (.5000) = 60° { 


— 60 c 


=fcn ■ 360°, (n = 0, 1, 2, . . .). 


3. Find the values of sin tan -1 3. 


We could solve this problem by finding the principal and second¬ 
ary values of tan- 1 3 with the aid of the Tables and again using the 
Tables to find the sine of each of those angles. Another method 
consists in writing 

a = tan -1 3, tan a = 3. 

Our problem may now be stated as follows: Find sin a, when it is 
given that tan a = 3. Problems of this sort have already been solved 
by the methods of page 29^and page 102. We thus obtain the 
result, sin tan -1 3 = dz3/VlO, the positive sign corresponding to 
the principal value Tan -1 3. 

4. Simplify the expressions sin sin -1 x } sin -1 sin x, and 
Sin -1 sin x. 


RADIAN MEASURE. INVERSE FUNCTIONS 139 


The expression sin sin- 1 # denotes the sine of an angle whose 
sine is x ; it can have but one meaning, 
sin sin -1 x = x. 

In the last two of the three given expressions, where x must 
denote an angle, the form of our answer will depend on whether x 
is given in degrees or radians; let us suppose here that the latter 
is the case. The function sin- 1 sin x is many-valued and its values, 
in radians, are as follows: 

sin -1 sin x = x, tt — x, 

or either of these values ±2 n?r, where n is any positive integer. 

Finally Sin -1 sin x is equal to x if x. is between — tt/2 and 
otherwise it is equal to the value of sin -1 sin x that is so situated. 


EXERCISES 


Find all the values of the following expressions without using 
the Tables; give results both in degrees and in radians: 


1 . (a) 


Sin -1 — 7 =; 

V2 


(c) Tan-M-l); 

2 . (a) Tan - 1 1; 

(c) Csc-M-D; 

3 . (a) sin-A; 

(c) cos - 1 (- 1 ); 

4 . (a) cot - 1 1; 

(c) sin -1 ^— >' 


(b) Cos- 1 -^; 

(d) Sec - 1 (- 2 ). 

(b) sin-‘(-A); 
(d) 

(b) tan -1 \/3; 
(d) cot -1 0 . 

(b) sec - 1 A; 

(d) tan -1 0 . 


Find all the values of the following expressions, using the 
Tables and giving results both in degrees and in radians: 

6 . (a) Sin- 1 .3000; (b) Tan" 1 .7125; 

(c) Cos " 1 (-.2300); (d) Cot " 1 (-2.002). 


140 


TRIGONOMETRY 


6 . (a) Cos" 1 .6000; 

(c) Tan - 1 (-1.256); 

7 . (a) sin -1 .7200; 

(c) sec " 1 (-2.035); 

8 . (a) tan -1 2.700; 

(c) cot ' 1 (-1.125); 


(b) Csc " 1 2.300; 

(d) Sin - 1 (-.0630). 
(b) cos- 1 .0325; 

(d) tan " 1 (-.0500). 
(b) sin- 1 .0750; 

(d) csc - 1 (-4.240). 


Find all the values of the following expressions without using 
the Tables: 


9 . (a) sin Sin -1 \; 

(c) sin Cos -1 i; 

10 . (a) cos Cos - 1 1; 

(c) cos Sin -1 f; 

11 . (a) sin Tan -1 f; 

(c) cos cot -1 (—V 2 ); 

12 . (a) tan Sin - 1 1; 

(c) cot sin -1 (-A); 


(b) sin sin -1 \) 

(d) sin cos -1 
(b) cos cos - 1 1 ; 

(d) cos sin -1 f. 

(b) tan Sec -1 (—{-); 
(d) sec sin -1 ( — f). 
(b) sin Sec -1 ( —f§); 
(d) cos tan -1 ( —f). 


Solve by using the Tables: 

13 . Exercises 11 (a), (b), (c), (d). 

14 . Exercises 12 (a), (b), (c), (d). 


Find the values of the following expressions without using the 
Tables: 

15 . sin (180° — Sin - 1 f). 

16 . cos (90° + Cos - 1 f). 

17 . tan (180° - Sin -1 4f). 

18 . cot [270° - Tan ' 1 (-§)]. 

19 . cos [180° +Cot " 1 (-2)]. 

20 . sin [270° - Cos " 1 (-i)]. 

*78. Graphs of inverse functions. Since the equations 
y = sin -1 a; and x = sin y are equivalent, their graphs 
are the same. In Figures 77, 78 (p. 97) we have given 
graphs for the equations y = sin x, y = tan x. If we inter¬ 
change x and y we obtain the graphs of sin -1 x and tan -1 x. 


RADIAN MEASURE. INVERSE FUNCTIONS 141 


In Figure 86 we show a portion of the graph of sin -1 x, 
leaving the construction of graphs of other inverse functions 
as an exercise. Here x is measured in radians. From P to 
Q we have the graph of Sin -1 x. The figure shows that x 


must lie between — 1 and + 1 if y = sin -1 x 
is to have a value; and that if x is so situ- 

1 

V 

r 

ated then the function sin -1 x has infinitely 
many values. 

7r 

\ 

* 79. Identities involving inverse func¬ 
tions. The following examples will show 
how to prove identities involving inverse 
functions by means of substitutions which 
permit us to express the problem in terms 

7r 

2 

7 

/ 

o 

of the ordinary functions. 

pi—.- 

7T 

Examples. — 1. Prove the identity 
sin 2 cos -1 x = 1 — x 2 . 

\ 

2 

To prove this formula, write 

_> 


cos -1 X = a, cos a = X. 

Fig. 86 


With this substitution our problem reduces to the following: Prove 
that sin* « = 1 - x* if x = cos a. In this form our identity is at 
once proved, since it reduces to sin 2 a = 1 - cos 2 <*. 


2. Prove the identity 

cos (2 sin -1 x) = 1 — 2 x 2 . 

Let 

sin -1 x - a, sin a = x. 

Then our identity reduces to 

cos 2 a = 1 - 2 z 2 = 1 — 2 sin 2 a, 

which is formula (2c) of page 116. 

If we express the identity we have just proved in the form 

2 sin -1 x = cos -1 (1-2 x 2 ), 

it is to be understood in the sense that each value of the inverse 






142 


TRIGONOMETRY 


function sin -1 x corresponds to some value of the inverse function 
cos -1 (1 — 2 x 2 ) by means of this formula, but a principal value of 
the one may not correspond to a principal value of the other. 

3. Prove the identity 

cos (Sin -1 x — Sin -1 y) = Vl — x 2 • Vl — y 2 + xy. 

Let 

Sin -1 a; = a, sin a = x (-90° ^ a ^ 90°), 

Sin -1 y = 0 , sin p = y (-90° ^ 0 ^ 90°). 

Then our formula becomes 

cos (a - 0 ) = Vl - sin 2 a ■ Vl - sin 2 0 + sin a sin 0 . 

Since a and 0 a re both be tween -90° and +90° , their cosines are 
positive and Vl — sin 2 a = cos a, Vl — sin 2 0 = cos 0 , so that 
the identity to be proved reduces to formula (4) of page 107. 

If sin - 1 x and sin - 1 y are substituted in the above formula for 
Sin -1 x and Sin -1 y, we must place the ± sign before the first term 
on the right. 


EXERCISES 


Prove the following formulas: 

1 . tan cot -1 x = - • 
x 


sec cos -1 x = 


3. sec 2 tan -1 £ = 1 + x 2 . 4. cos 2 esc -1 z = 1 — 

x 

5 . cos Sin -1 x = Vl — x 2 . 6 . sin Cos -1 x = Vl — x 2 . 

7 . cos (2 cos -1 x) = 2 x 2 — 1 . 

2 x 

8 . tan (2 tan -1 x) = ± _ x 2 * 

9 . tan (tan -1 x — tan -1 y) = r—- 

i -f- xy 

10. sin (Sin -1 x + Sin -1 y) = x Vl — y 2 + y Vl — x 2 . 

11. sin (sin -1 x — cos -1 y) = xy ± V (1 — x 2 ) (1 — y 2 ). 















RADIAN MEASURE. INVERSE FUNCTIONS 143 


12. sin (J cos -1 x) = d= y / 1 2 

13. Sin -1 x — — Cos -1 x. 


14. Tan -1 x = - — Cot -1 x. 

15. Tan- 1 a = Sin- 1 . a - ■ 

Vl +a 2 


Prove that the following formulas are true in the sense ex¬ 
plained in Example 2, § 79: 

, , , i i x , x + y 

16. tan -1 x + tan -1 u = tan -1 - - — • 

1 ~ xy _ _ 

17. cos -1 x — cos -1 y = cos -1 {xy ± Vl — x 2 Vl — y 2 ). 

18. J cos -1 a = tan -1 

Prove that the following equations are true: 



19. Tan- 1 Tan-4 = j. 

20. Sin" 1 (-*) - Sin -1 

21. 2 Tan -1 f = Tan -1 ^ 2 -. 

22. Sin -1 1 — Sin -1 \ = Sin -1 ^ • 

23. Sin" 1 f + Sin" 1 ff = Cos" 1 (-if). 











CHAPTER VII 

TRIGONOMETRIC EQUATIONS 


80. Definitions. Equations containing trigonometric 
functions of unknown angles are called trigonometric equa¬ 
tions . Examples of such equations with one unknown are 

(a) cos x = 1, (b) cos 2 x + sin x = 1, (c) x = tan x. 

We may also have simultaneous trigonometric equations 
with two unknowns. Thus if (a, b) are the rectangular co¬ 
ordinates of a point we find its polar coordinates (§§ 13, 14, 
pp. 13-16) by solving for r and 6 the pair of equations 

r cos 6 = a, r sin 6 = b. 

We may simplify a trigonometric equation by the ordinary 
algebraic processes, such as clearing of fractions, transposing 
terms, and taking a root or a power of both sides (with the 
precautions explained in algebras). We may also use 
trigonometric transformations. Thus in example (b) of the 
preceding paragraph we would replace cos 2 x by 1 — 2 sin 2 x 
in order to reduce the equation to one in a single trigonometric 
function of x. 

A solution of an equation in one unknown, x, is a value of 
x for which the equation holds true. Thus for the equation 
cos x = 0, solutions (in radians) are 

7r 7T 3 7T 5 7T 

x ~ 2 ’ ~~ 2 ’ ~2 ’ ~ 2 ~’ ' 

Many trigonometric equations are like this one in possessing 
an infinite number of solutions, whereas algebraic equations 
in one unknown which are not identities have only a finite 
number of solutions. Although cos x = 0 has an infinite 
144 


TRIGONOMETRIC EQUATIONS 


145 


number of solutions, it is not an identity since it is not true 
for all values of x in any interval. Throughout the present 
chapter we shall consider only equations that are not iden¬ 
tities. 

81. Simple examples. The equations 

sin x = a, cos x = a, tan x = a, 

cot x = a, sec x = a, esc x = a, 

have already been discussed, particularly in §77 (p. 137). 
Here a is a given number, and x is found as an inverse trig¬ 
onometric function of a. We obtain a principal and a sec¬ 
ondary value of x, using the Tables if necessary, and all other 
solutions are derived from these by adding or subtracting 
multiples of 360° or 2 r radians. Certain other equations 
are easily reduced to this form. 

Examples. — 1. Find all the solutions of sin x = cos x. 

Divide both sides by cos x, first noting that cos x cannot be zero 
if x is to be a solution (Why?). The equation becomes 

tan x = 1, 

and our solution is x = tan -1 1, or 

x = 45° ± n • 360°, 225° ± n • 360°, 

where n is zero or any positive integer. 

If we had started by squaring both sides, then using the relation 
cos 2 x = 1 — sin 2 x, we could have proceeded as follows: 

sin 2 x = cos 2 x, 
sin 2 x = 1 — sin 2 x, 

2 sin 2 x = 1, 
sin z = =b 

This seems to give additional solutions x = 135°, 315°, • • • , 
but in algebra we learn that squaring both sides of an equation, or 
multiplying both sides by an expression containing the unknown, 
is allowable only if we test all solutions of the new equation by sub¬ 
stitution in the original equation, retaining only those that satisfy 


146 


TRIGONOMETRY 


the latter. In the example we are considering, this test would cause 
us to reject x = 135°, 315°, • • • , and retain those noted in the 
preceding paragraph. 

2. Find in radians all the solutions of 2 sin 2 x = 1 that 
lie between 0 and 2 t. 

We have 

• 1 

sm 2 x = 2 ’ 

2 x = | + n • 2 tt, + n ■ 2 tt, 

TT 5 TT 

X = J 2 + W7T, -J 2 + U7r > 

and the solutions required are obtained by taking n = 0 and n = 1 . 
This gives 

TT 5 7T 13 TT 17 TT 

x ~ 12 f 12 ’ 'T 2 " , nr* 

3. Find in degrees and minutes all positive solutions less 
than 180° of 

sin 2 x — 4 cos 2 x + 2 = 0. 

We use the relation sin 2 x = 1 — cos 2 z and proceed as follows: 

sin 2 x — 4 cos 2 # + 2 =0, 

1 — cos 2 x — 4 cos 2 z + 2 =0, 

—5 cos 2 x = —3, 

3 

cos 2 x = 5 = .6000, 
cos x = ± V.6000 = ± .7746. 

Hence, from Table II, one solution is 39° 14'; the other solution 
is 180° - 39° 14' = 140° 46'. 

82. Factorable equations. As in algebra, we may solve 
an equation in which one side is the product of two or more 
expressions and the other side is zero, by equating to zero 
each factor that contains an unknown. Sometimes one or 


TRIGONOMETRIC EQUATIONS 


147 


more of the identities of Chapter V will serve to reduce an 
equation to factorable form. 

Examples. — 1. Find all solutions of 2 sin 2 d = sin 6 such 
that 0° ^ 0 ^ 180°. 

We can write this equation in the forms 

2 sin 2 9 — sin 9 = 0, 
sin 9 (2 sin 9 — 1 ) =0. 

Both factors give solutions. 

It would have been a mistake to cancel the common factor sin 9 
in the original equation and solve only the equation 2 sin 9 = 1. 
The student must be on his guard against thus throwing away 
solutions. Our problem is now to solve the two equations 

sin 9 = 0, 2 sin 9 — 1 = 0. 

The solutions required are 

9 = 0°, 180°, 30°, 150°. 

2 . Find in radians all positive solutions less than x of 

cos x + cos 2 x + cos 3 x = 0. 

We begin by using formula (7) of page 122 in order to change the 
sum cos x + cos 3 x into a product: 

cos x + cos 3 x = 2 cos 2 x cos x. 

The given equation is then solved as follows 

cos x + cos 2 x + cos 3 x 
2 cos 2 x cos x + cos 2 x 
cos 2 x (2 cos x + 1 ) 
cos 2 x = 0 , 2 cos x + 1 

7r 3 7T 2 7T 

X= V T’ T 

83. Equations reducible to quadratic form. An equation 
such as sin 2 x — 3 sin x + 2 = 0, which contains but one 
trigonometric function of the unknown and is of second 


= 0 , 
= 0 , 
= 0 , 
= 0 , 


148 


TRIGONOMETRY 


degree in that function, can first be solved for the function 
by factoring or by other algebraic means. The problem is 
thus reduced to that of solving simple equations of the type 
sin x = a, discussed in § 81. 

For example, the equation 

sin 2 x — 3 sin z+2=0 
can be written in factored form 

(sin x — 2) (sin x — 1) =0. 

When each factor is put equal to zero we note that sin x — 2 = 0 
has no solutions, while sin x — 1 =0 gives 

x = 90° ± n 360°. 

In many cases an equation may be reduced by algebraic 
or trigonometric transformations to the type discussed in the 
preceding paragraph. 

Examples. — 1. Find all solutions of 

sin 0 + 2 cos 6 = 2 
such that 0° ^ 6 ^ 180°. 

We reduce this equation to quadratic form by transposing the 
term 2 cos 0 to the right side, squaring both sides, and replacing 
sin 2 9 by 1 — cos 2 9 . We have 

sin 9 + 2 cos 9 = 2 , 
sin 9 = 2 (1 — cos 9 ), 
sin 2 9 = 4 (1 — cos 0) 2 , 

1 — cos 2 0=4—8 cos 0+4 cos 2 0, 

5 cos 2 0 — 8 cos 0 + 3 =0, 

(cos 0 — 1) (5 cos 0—3) =0, 
cos 0 = 1, cos 0 = .6, 

0 = 0°, 53° 8'. 

We must, however, test both these values in the original equation 
since we squared both sides at the second step. It will be found 
that the two values for 0 are actually solutions. 


TRIGONOMETRIC EQUATIONS 


149 


2. Find all solutions of 

cos 2 0 + 6 cos 2 ^ — 4 = 0 

that lie between —90° and +90°. 

This equation is reduced to a quadratic in cos 0 by using formulas 
given in Chapter V. We proceed as follows: 

Q 

cos 2^+6 cos 2 ^ — 4 = 0, 

2cos‘0 — i +6 !±®“?_4 =0> 

2 cos 2 0+3 cos 0—2 =0, 

(2 cos 0 — 1) (cos 0+2) =0, 
cos 6 — 2 } 

0 = 60°, -60°. 

The factor cos 0 + 2 yields no solution, since cos 0 cannot equal —2. 


EXERCISES 

Find, both in degrees and in radians, all solutions of the 
following equations. 

1 . 2 cos 0 + 1 = 0. 2. 1 + 2 sin 0 = 0. 


3. tan 2 x - 6 = 0. 

5. sin 0 = 2 cos 0. 

For the following equations 
0 < x ^ 360°. 

7. sin 2 x = 2 sin x. 

9. sin 2 x — cos x = 0. 

11 . cos 4 x — cos 2 x = 0. 

13. sin (x + 60°) = sin x. 

15. sin x — sin 2 x + sin 3: 

16. cos 3 x + sin 2 x — cos 

17. tan 2 x + 2 = 3 tan x. 

19. esc x — sin x = + 

21. cos2z+cos:r+l =0. 


4. 6 tarn 0—1=0. 

6. cos 0 = 2 sin 0. 

find all solutions such that 

8. tan 2 x = 2 tan x. 

10. since + cos 2x = 1. 

12. sin 3 x + sin x = 0. 

14. cos (x — 30°) + cos x = 0. 
= 0 . 

= 0 . 

18. 2 cos 2 x + 3 = 5 cos x. 

20. sec x + 2 cos x = 3. 

22. cos2cc — 2 sin cc+| = 0. 



150 


TRIGONOMETRY 


For the following equations find all solutions 0 such that 
-90° ^ 0 ^ 90°. 

23. sin 0 + sin 3 0 = cos 0 — cos 3 0. 

24. sin 4 0 — sin 2 0 = cos 3 0. 

25. sin (0 — 60°) — sin (0 + 60°) + ^ V3 = 0. 

26. sec (0 + 120°) + sec (0 - 120°) = 2 cos0. 

27. tan 2 0 + 4 sin 2 0 = 3. 28. cot 2 0 = tan 0 - cot 0. 

29 2cos^ = —esc0—cot0. 30. sin 4 0 + cos 4 0 = 

2 z 

31. 2cos0 — sin0 = 1. 32. 2sin0.+ cos0 = 2. 

33. 8sin0 + cos0 = 7. 34. 4sin0 - 7cos0 = 1. 

35. cos 3 0 = 4 cos 2 0. 36. sin 3 0 + 4 sin 2 0 = 0. 

*84. The type a sin x + b cos x = c. If we make the 
substitutions 

(1) a — r cos a, h = r sin a, 

the left side of our equation becomes 

r cos a sin x + r sin a cos x = r (sin x cos a + cos £ sin a) 

— r sin ( x + a). 

We now proceed as follows: 

r sin (x + a) = c, 

^ sin (x + a) = C -> 

• -i c 

x — sm 1 — a. 
r 

In order to express r and a in terms of a and h we square 
and add equations (1) obtaining 

r = Va 2 + b 2 , 
b 

Va 2 -h b 2 


r 2 = a 2 + b 2 , 
a 


cos a = 


Va 2 + b 2 







TRIGONOMETRIC EQUATIONS 151 


These equations determine values of r and a that are to be 
used in (2). 

Example. — Solve the equation 4 sin x — 7 cos x = 1 (Ex. 
34, p. 150). 


Here we have 

r = V4 2 + (-7) 2 = V65, 


COS a 


4 

V65’ 


sin a 


-7 

V65* 


From the last two equations we see that a terminates in the fourth 
quadrant. From (2), 


= sin ~‘4l - Sin_l vi 

= sin_1 ^3 + sin "^!' 


Hence 


= Sin~ 


V65 


+ Sin - 


L -4= ± n 360° 
V65 


180 ° - Sin ~‘ Wb + Sin_1 4s ± n 360 ° 

= 67° 23' =fc n 360° or 233° 8 ' ± n 360°. 


^85. Approximate solutions. Many equations cannot be 
solved by the methods of the preceding sections. We can, 
however, often obtain approximate solutions, either graphi¬ 
cally or with the aid of the Tables. 

For example, let us find solutions of the equation 

2 x — tan x = 0 


such that - § < * < f > x being measured in radians - If 

we draw the graphs of the equations y = 2 * and y = tan * 
(see Fig 87 (p. 152) where distances on the z-axis represent 
radians) the abscissas of their points of intersection will 
furnish solutions. This follows from the fact that it 



152 


TRIGONOMETRY 


(xi, yi) is a point of intersection, we shall have yi = 2 xi 
and yi = tan x\, so that 2 x\ — tan Xi = yi — yi = 0. The fig¬ 
ure shows that there are three such 
intersection points in the interval we 
are considering. The corresponding 
values of x } which are solutions of 
our equation, can be measured and 
will be found to be 
x = 0 

and the two approximate values 
Xi = 1.15, x 2 = —X\ = —1.15. 

We could obtain x\ (and x 2 , which 
is equal to —Xi) more exactly by 
using the Tables. For we have 
tan xi = 2 Xi, thus, we are to find an 
angle whose tangent is twice its radian measure. By com¬ 
paring the radian column of Table II with the tangent col¬ 
umn we see that tan x is less than 2 x until x becomes greater 
than 1.1636. We have, from the Tables, 

if x = 1.1636, then 2x — tana: = +.009; 
if x = 1.1665, then 2 x — tana: = —.004. 

The principle of proportional parts would place Xi, for which 
2x — tana: equals zero, 4/13 of the way from 1.1665 to 
1.1636. This gives 

xi = 1.1665 - X 29) = 1.1656. 

EXERCISES 

Solve for all values of 0 such that 0° ^ 0 ^ 180°: 

1. sin 9 — 8 cos 6 = 7. 2. 8 sin 6 + cos 6 = 7. 

3. 3sin0 + 4cos0 = 3. 4. 5 sin 6 — 12 cos 6 = 9. 

5. 5 sin 0 — 12 cos d = 13. 6. 3 sin 0 — 4 cos 0 = 5. 


Y 







TRIGONOMETRIC EQUATIONS 


153 


In the following equations 7 to 12, x is measured in radians. 


Find all solutions such that 

< 

2 = 

x - 2* 

7. 

3 x = 2 tan x. 

8. 

4 x = tan x. 

9. 

x + 1 = tan x. 

10. 

3 x — 1 = tan x. 

11. 

x — | sin x = £. 

12. 

2 x = cos x + ^ • 

13. 

In a circle whose radius is 

10 in., how long is a chord 


that subtends a segment of area 100 sq. in. (§ 74, p. 132)? 

14. In a circle whose center is at 0, a sector AOB has 
twice the area of the triangle AOB. Find the angle AOB. 

15. An arc of a circle (greater than a semicircumference) 
is twice as long as its chord. Find the subtended angle. 

16. A segment of a circle has an area equal to one-fourth 
that of the circle. What is the ratio of its arc to the circum¬ 
ference? 


CHAPTER VIII 

LOGARITHMS 


When a computation requires a long multiplication or 
division, the raising of a number to a power, the extraction 
of a root, or a succession of such operations, the work may be 
shortened and the probability of an important error lessened 
by the use of logarithms. 

The theory of logarithms rests directly on the theory of 
exponents. We therefore start our discussion of the former 
by a brief review of the latter. 

86. Exponents. The reader will recall that by definition 

10 2 = 10X10, 10 3 = 10X10X10, 10 5 = 10X10X10X10X10. 


It follows that 

10 2 X 10 3 = 10 5 , 



IT 2 = J_ 
10 5 10 3 * 


These three equations are examples of the general laws of 
algebra contained in the formulas 


( 1 ) 

( 2 ) 


a m X a n = a m+n ; 



•n — 


1 

a n -m‘ 


Here and throughout this chapter we shall assume that the 
base a is positive; the exponents m and n are any real numbers. 
It follows from the preceding paragraph that 

(10 3 ) 2 = 10 3 X 10 3 = 10 6 ; (10 5 ) 3 = 10 15 . 

These are special cases of the general law, 


LOGARITHMS 


155 


We next recall the use of fractional exponents. The 
definition of a fractional power is to be so made that laws 

(1) , (2), and (3) hold. By (3) we must have (10 1/2 ) 2 = 10; 
hence 10 1 / 2 must be a square root of 10. The general defi¬ 
nition is as follows: If r is a positive integer, then (a) 1/f is 
the positive rth root of a. This is written 

(4) (a) 1 /' = </a. 

Again by (3) we have (10 3 ) 1/2 = 10 3/2 and (10 1/2 ) 3 = 10 3/2 ; 
hence 10 3 / 2 may be expressed, by virtue of (4), in either of 
the forms (10 1/2 ) 3 = (VTO) 3 or (10 3 ) 1 / 2 = VlO 3 . The cor¬ 
responding general formula is 

(5) aP/' = </aP = (S/a)K 

The definition of a negative power is guided by equations 

(2) . Taking m = 0 we have 


A definition of the zero power may be arrived at as follows. 
By (2), 10 3 /10 3 = 10°; but a number divided by itself gives 
1, so that 10 3 /10 3 = 1; hence 10° = 1. The general defi¬ 
nition is 

( 7 ) a 0 = 1 . 

The definition of an irrational power is too complicated 
to explain here in detail. It will suffice to say that if m is 
an irrational number which is closely approximated by a 
rational number m', then a m is closely approximated by a m '. 
Thus, since \/2 = 1.414 • • • = 1414/1000 approximately, 
we have a^ = a 1414 / 1000 approximately. 

Example. — Let us find the values of a few powers of 10. We 
shall take a set of exponents of which the first is 1, and each there¬ 
after is equal to half of its predecessor. Since by (5) 
aP/ 2 = V afi, 


156 


TRIGONOMETRY 


each of our numbers will be the square root of the one before it. 
We have 

10 1 = 10 , 

10- 5 = 10 1 / 2 = Vlo = 3.1623, 

10- 25 = 10 1 / 4 = (10 1 / 2 ) 4 / 2 = Vl0^ = V3.1623 = 1.7783, 

lO.m = loi/s = Vl.7783 = 1.3335, 

10-0625 = loi/is = VL3335 = 1.1548, 

10-03125 = 101/32 = V0548 = 1.0746. 

The values in the right members are correct to five significant 
figures. 

If from the last result, 

10-03125 = 1.0746, 

we form successive powers by multiplying each member by itself 
repeatedly, we get 

10.06250 = 1 . 1548 , 

10-09375 = 1.2409, 

10.12500 = 1.3335, 

10-15625 = 1.4330, 

and so on. If we continue the process, the thirty-second equation 
will be 10 1 = 10. We thus have 31 numbers between 1 and 10 
expressed as powers of 10. We note that as the numbers on the 
right increase in these equations the exponents in the left members 
also increase. Also we remark that the exponents all lie between 
0 and 1, and the numbers on the right between 1 and 10. 

87. Expressing numbers as powers of 10. A very im¬ 
portant fact at the basis of the theory of logarithms is con¬ 
tained in the following statement : 

Theorem. Every positive number can be expressed as a 
power of 10, and there is only one power of 10 which will yield 
a given number. 

A complete proof of this statement cannot be given here. The 
theorem is made very plausible, however, as follows. In the ex- 






LOGARITHMS 


157 


ample in § 86, we see how 31 numbers between 1 and 10 are ex¬ 
pressed as powers of 10. If we carry out the extraction of square 
roots in that example for five more steps we will have the value of 
10 1 / 1024 . From this we obtain, on taking successive powers in the 
manner indicated in the example, 1023 numbers (instead of 31) 
between 1 and 10 expressed as powers of 10. If we carry out the 
extraction of square roots to a total of 20 steps and then form 
successive powers, we have over a million numbers between 1 and 
10 expressed as powers of 10. It thus becomes apparent that any 
number between 1 and 10 can at least be very closely approximated 
by a power of 10. And since when the numbers increase the corre¬ 
sponding exponents increase, there will be only one exponent which 
will yield a given number. 

As for numbers not between 1 and 10, consider first two typical 
examples. We may write 

1154.8 = 1.1548 X 10 3 = 10- 0625 X 10 3 = 10 3 - 0625 
.011548 = 1.1548 ^ 10 2 = 10- 0625 X 10~ 2 = 10- 0625 - 2 . 


Since any positive number can be expressed similarly as the product 
of an integral power of 10 and a number between 1 and 10, it can 
likewise be expressed as a power of 10. 


88. Definition of the logarithm of a number. If a number 
N is expressed as a power of 10, 

N = 10*, 

then the exponent, x, is called the logarithm of N (to the base 
10); in symbols we write, 

log N = x. 

Thus by definition 

10 log N = N. 


An immediate consequence of the theorem of the preceding 
section is the following: 

Theorem. Every positive number N has one and only one 
logarithm (to the base 10). 


158 


TRIGONOMETRY 


Another consequence of the discussion in the preceding 
section is that if one number is greater than another its log¬ 
arithm is also greater. 

No power of 10 yields a negative number; hence negative 
numbers do not have logarithms. 

As examples of logarithms, we may write the following 
pairs of equivalent statements: 


10 = 10 1 , 

log 10 = 1 ; 

100 = 10 2 , 

log 100 = 2; 

1000 = 10 3 , 

log 1000 = 3; 

1 = 10°, 

log 1 = 0; 

.1 = 10- 1 , 

log -l = -1; 

.01 = 10- 2 , 

log .01 = -2. 

Similarly the final equations of the example in 

may be written in the equivalent forms, 


log 1.0746 = .03125, 
log 1.1548 = .06250, 
log 1.2409 = .09375, 
log 1.3335 = .12500, 
log 1.4330 = .15625. 


Note. It is sometimes useful to replace 10 by some other 
number in the definition of a logarithm. The more general defini¬ 
tion is, if 

N = a x 

then x is the logarithm of N to the base a, and we write 
log a N = x. 

For computational purposes the base 10 is most convenient. For 
theoretical purposes in higher mathematics a base called e , where 

e = 2.71828 • • • , 

is simplest to use. Logarithms to the base 10 are called common 
logarithms; to the base e natural logarithms. 


LOGARITHMS 


159 


EXERCISES 


Find values of the following: 

1 . 3 2 X 3 3 ; (3 2 ) 3 ; (Z 2 ) 1 ' 2 ; 

2. 2 3 X 2 2 ; (2 3 ) 2 ; (2 3 )V3 ; 

3. 10* 375 . {Hint. 10* 375 = 

4. 10- 625 . 

6. log 3.1623. 

8. log 11.548. 

10. log .11548. 

12. log .001. 


( 8 1/3 ) 2 ; 8 " 2 / 3 . 

(9 1/2 ) 3 ; 9 _3/2 . 

10- 25 X 10- 125 ; see § 86.) 

5. log 1.7783. 

7. log 1154.8. (See §87.) 
9. log .011548. 

11 . log 1,000,000. 

13. log .0001. 


89. Fundamental laws of logarithms. The great useful¬ 
ness of logarithms arises from the following fundamental laws, 
which are proved below: 

I. The logarithm of the 'product of two numbers equals the 
sum of the logarithms of the factors. Stated in symbols, 

(1) log MN = log M + log N. 

II. The logarithm of the quotient of two numbers equals the 
logarithm of the dividend minus the logarithm of the divisor. 
Symbolically, 

M 

(2) log = log M - log N. 

III. The logarithm of the nth power of a number equals n 
times the logarithm of the number. That is, 

(3) log M n = n log M. 

IV. The logarithm of the rth root of a number is one rth of 
the logarithm of the number. Symbolically, 

(4) log v^M = jlog M. 

The proofs of these theorems are as follows: 


160 


TRIGONOMETRY 


By definition 

(5) M = 10 IogM , N = 10 Iog ^, 

and 

MN = 10 Iog MN . 

But from (5) we have, by the first rule of exponents, (1), 
§ 86 (p. 154), 

MN — io logM + logi L 


Since there is only one power of 10 which equals MN, we 
therefore have 

log MN = log M + log N, 

which is Law I. 

Similarly, by definition, 


M =10 

N 


log ; 


But from (5) and the second rule of exponents, (2), § 86 
(p. 154), we have 


Hence 


M 10 Iog M 
~N~10Wn 


^Qlog M— log N' 


log = log M - log N, 


which is Law II. 

To prove the third law, we note first that by definition 

M n = io Io s M% . 

And secondly, from (5) and the third rule of exponents, 
(3), § 86 (p. 154), we have 

M n = (l0 logM )” = i0 wlogM . 



LOGARITHMS 


161 


Hence 

log M n = n log M, 

which is Law III. 

The fourth law follows from the third since, by (4), § 86 
(P- 155), 

y/M = M 1/r . 

For we have 

log </M = log M l " = - log M. 

r 

Note. The preceding laws are true whatever base of logarithms 
is used. To prove them for a base a, we simply replace 10 by a 
throughout the argument. 

Example. — A very simple application of the first law is the 
following. We have (p. 158) 

log 10 = 1, log 1.433 = .15625. 

Since 

14.33 = 10 X 1.433, 

it follows that 

log 14.33 = log 10 + log 1.433 = 1.15625. 

Similarly 

log 143.3 = log 100 + log 1.433 = 2.15625, 

log .1433 = log .1 + log 1.433 = -1 + .15625, 

log .01433 = log .01, + log 1.433 = -2 + .15625, 

log .001433 = log .001 + log 1.433 = -3 + .15625. 


EXERCISES 

Find the values of the following logarithms hy use 
values given on page 158: 


1 . 


log 10.746, 
log 107.46, 
log 1074.6, 
log 10746, 
log 107460. 


2 . 


log 11.548, 
log 115.48, 
log 1154.8, 
log 11548, 
log 115480. 


3. 


of the 


log 12.409, 
log 124.09, 
log 1240.9, 
log 12409, 
log 124090. 


162 


TRIGONOMETRY 


4 . log .10746, 
log .010746, 
log .0010746, 
log .00010746. 

7 . (a) log 10.746 3 ; 

8. (a) log 1.1548 10 ; 


5 . log .11548, 
log .011548, 
log .0011548, 
log .00011548, 


6. log .12409, 
log .0124092, 
log .00124092, 
log .000124092. 


(b) log V 107.46T 
(b) log 'V / 115.48. 


90. Characteristic and mantissa. In this section we shall 
understand that all numbers are written in decimal form. 

As indicated on page 158, the logarithms of the numbers 10, 
100, 1000, ... are the positive integers 1, 2, 3, . . . ; the 
logarithm of 1 is 0; and the logarithms of .1, .01, .001, . . . 
are the negative integers —1, —2, —3, . . . . The loga¬ 
rithm of any other positive number can be expressed as the 
sum of an integral part and a positive decimal part. The 
integral part is called the characteristic, the decimal part the 
mantissa of the logarithm of the number. 

In the example at the end of the last section we had 

log 1.433 = 0.15625, log .1433 = -1 + .15625, 

log 14.33 = 1.15625, log .01433 = -2 + .15625, 

log 143.3 = 2.15625, log .001433 = -3 + .15625. 


The characteristics are 0, 1, 2 in the first column, —1, —2, 
— 3 in the second. The mantissas are all the same, .15625. 

The logarithm of any number between 1 and 10 lies be¬ 
tween log 1 and log 10, that is, between 0 and 1. Hence, 
the characteristic of the logarithm of any number between 1 and 
10 is 0. 

To get a general rule for finding the characteristic let us 
first recall from arithmetic that by units’ place in a number 
we mean the first place to the left of the decimal point when 
the number is written in decimal notation. Thus for each 
of the numbers 4.2, 34, and 604.71, the digit 4 is in units’ 
place. 

Suppose now that, for a given number N, in going from the 




LOGARITHMS 


163 


first significant figure to units’ place we move 4 places to the 
right; then the number can be expressed as 10 4 N' where N' 
is a number between 1 and 10. Thus 14330 = 10 4 X 1.433. 
Hence 

log N = log 10 4 + log N' = 4 + log N'; 

the characteristic of log N is 4. 

Suppose next that in going from the first significant figure 
of N to units’ place we move 4 places to the left; then the 
number can be expressed as 10 -4 N', where N' is between 1 
and 10. Thus .0001433 = 10~ 4 X 1.433. Hence 

log N = log 10 -4 + log N f = —4 + log N'; 

the characteristic of log N is —4. 

The reasoning in the last two paragraphs is obviously 
general in character. If we replace 4 by A; we get the fol¬ 
lowing rule: 

To find the characteristic of log N, first find how far it is 
from the first significant figure of N to units’ place. If it is 

k places to the right, the characteristic is k, 
k places to the left, the characteristic is —k. 

Thus the characteristic of log 9.3 is 0; of log 93,000,000 
is 7; of log .123 is —1; and of log .000005 is —6. 

Another rule sometimes used in finding the characteristic is as 
follows: If in a number N there are n digits to the left of the decimal 
place, the characteristic of log N is n - 1. If the number N is less 
than 1 the characteristic is negative and one greater than the num¬ 
ber of zeros between the decimal point and the first significant 
figure in N. 

From the preceding paragraphs we see that the mantissa 
of log N is log N' where N' is the number between 1 and 10 
which is obtained from N by merely shifting the decimal 
point to the proper place. Hence the mantissa depends only 


164 


TRIGONOMETRY 


on the succession of digits in N, and not at all on the 'position 
of the decimal point. Accordingly the decimal point may be 
ignored when one looks for the mantissa. The mantissa is 
found from a table of logarithms, as explained in the next 
section. 

When the characteristic is negative care must be taken in 
writing the logarithm. Thus it would be a mistake to write 

log .1433 = -1.15625, 

for the number in the right member equals —1 — .15625, 
and not the correct value — 1 + .15625. One commonly 
used way of writing the logarithm is 1.15625, it being under¬ 
stood that only the characteristic is affected by the negative 
sign. Another method is to use such relations as 

_1 = 9 - 10 or -1 = 19 - 20, 

and write 

log .1433 = 9.15625 - 10 = 19.15625 - 20. 

In this book we shall adopt the latter system, in which the 
negative characteristic is expressed as a positive integer 
minus a multiple of 10. 

Note. By reviewing this section it may be seen that if another 
base of logarithms than 10 were used we would not have such simple 
rules for finding the characteristic and mantissa. It is because of 
this relative simplicity that the base 10 is generally used in com¬ 
putation. 

91. Finding logarithms from a table. A table of loga¬ 
rithms gives approximate values of the mantissas for a set 
of numbers. Thus in Table III the mantissas are given 
correct to four decimal places for the integers from 100 to 
999. In Table VII they are given to five places for the 
integers 1 to 100 and 1000 to 10009. The direct use of the 
Tables is illustrated in the following examples. 

Examples. — 1. To find log 320 to four places. 



LOGARITHMS 


165 


From the rule we find that the characteristic is 2. For the 
mantissa turn to Table III. We go down the column headed N 
to the number 32, across the row to the column headed 0 and find 
5051. When the decimal point, which is omitted in the Table for 
simplicity in printing, is placed ahead of the first 5, this is the 
mantissa. Hence 

log 320 = 2.5051 to four places. 

2. To find log 325 to four places. 

In this case go across in the row 32 to the column headed 5 and 
find 119. The first figure of log 320 which occurs at the beginning 
of the row 32 in column 0 is understood to precede this, so that 
the mantissa is .5119; hence 

log 325 = 2.5119. 

3. To find log .507 to four places. 

To go from the first significant figure, 5, to units’ place we move 
one place to the left; hence the characteristic is —1. In Table III, 
in row 50 go across to column 7, and find *050; this is not to be 
preceded by the first figure, 6, in log 500; the * calls attention to 
a change, and we are to take the first figure, 7, of logarithms in the 
next row. Thus the mantissa is .7050, and we have 

log .507 = 9.7050 - 10. 


4. To find log .06378 to four places. 


We may form the little table to the 

N 

log N 

right by reference to Table III. The 

637 

8041 

required logarithm is .8 of the way from 

637.8 


log 637 toward log 638. Hence we must 
add .8 of the difference 8048 — 8041 as 

638 

8048 


a correction to 8041; the correction is therefore .8 X 7 = 5.6 = 6 
approximately. The same correction could be found in the mar¬ 
ginal table on the right in row 63 and column 8. We add the 
correction and put in the decimal point to get the mantissa. The 
characteristic being —2, we have the result 

log .06378 = 8.8047 - 10. 


166 


TRIGONOMETRY 


5. To find log 4680 to five places. 

Turn to Table VII (p. 8i). In column N go down to row 468 and 
in column 0 find 67025. The decimal point is to be placed before 
the 6 to give the mantissa. Since the characteristic is 3 we have 

log 4680 = 3.67025. 

6. To find log .4691 to five places. 

On page 81 in row 469 and column 1 we find 127. This is to be 
preceded by the first two digits 67 of log 4680, giving 67127. Since 
the characteristic is —1, we have the result 

log .4691 = 9.67127 - 10. 

7. To find log .04679 to five places. 

On page 81, in row 467 and column 9 we find *015. If it were not 
for the * we w^ould place the two digits 66 of column 0 before these 
three, but the * indicates a change to 67 which occurs in the follow¬ 
ing row. The characteristic being —2, w r e have 

log .04679 = 8.67015 - 10. 

8. To find log 15897 to five places. 

From page 75 of the Tables we form the little table shown to the 
right. We must interpolate. The re¬ 
quired logarithm is .7 of the way from N log N 

20112 to 20140. Hence we must add 1589.0 20112 

to the former the correction found by 1589.7 

taking .7 of the difference 20140 — 1590.0 20140 

20112 = 28, that is, .7 X 28 = 19.6 = 

20 approximately. This correction could be found by looking in 
the proportional parts (Prop. Pts.) table on the margin of page 75, 
in the Tables, in column 28 and row 7, where we find 19.6. The 
interpolated value of log N is therefore 20112 + 20 = 20132. 
Putting in the decimal point, and observing that the characteristic 
is 4, we have 


log 15897 = 4.20132. 


LOGARITHMS 


167 


EXERCISES 

Find the characteristic of the logarithm of each of the fol¬ 
lowing numbers: 

1. (a) 2.468; (b) 2468; (c) .2468; (d) .0002; 

(e) 4.2 X 10 -6 . 

2. (a) 35.72; (b) 35720; (c) .0357; (d) .0010; 

(e) 5.6 X 10- 3 . 

3. (a) 365.1; (b) 25000; (c) .00254; (d) .00003; 

(e) 4.9 X 10- 9 . 

4. (a) 17; (b) 231.5; (c) .000444; (d) .31313; 

(e) 2.7 X 10- 16 . 

Find the logarithm of each of the following numbers by use 
Df Table III: 

6. (a) 36.2; (b) .0961. 6. (a) 481; (b) .00629. 

7. (a) 946; (b) .9468. 8. (a) 85300; (b) .08532. 

9. (a) .002561; (b) 3194. 10. (a) 798.2; (b) .0006398. 


Find the logarithms of each of the following numbers by use 


of Table VII: 

11. (a) 174.4; (b) .8928. 

13. (a) 2189; (b) .06769. 

15. (a) 37377; (b) .0089163 

17. (a) 57.546; (b) .40773. 


12 . (a) 7477; (b) .01905. 

14. (a) 6.459; (b) .002639. 

16. (a) 145.58; (b) .74177. 

18. (a) 45.709; (b) .097736. 


Correct the following: 

19. (a) log 9099 = .9589; 

(b) log .3382 = 9.5291; 

(c) log .004175 = 8.6206 - 10. 

20. (a) log 478.85 = 2.67019; 

(b) log .57598 = 1.76040; 

(c) log .0033885 = 7.52000. 


92. Finding a number whose logarithm is given. If the 

logarithm of a number is given and the number is required, 


168 


TRIGONOMETRY 


the steps of the preceding section are reversed, as illustrated 
in the following examples. 

Examples. — 1. Given log N = 1.9258. To find N. 

We look in the four-place logarithm table for the mantissa .9258. 
On page 11 we find the corresponding number 8430, the final zero 
indicating that no interpolation is necessary and that the number 
differs from 8430 by very little, — less than 1. Since the charac¬ 
teristic is 1, units’ place is one place to the right of the first significant 
figure. Hence 

N = 84.30. 


2. Given log N = 5.5011. To find N. 

The mantissa .5011 is found in row 31 and column 7; it corre¬ 
sponds to the number 3170. Since the characteristic is 5, units 
place is 5 places to the right of the 3. Hence 

N = 317000 to four significant figures. 


'•[ 


N 

log N 

6420 

80751 


8080J 

6430 

8082 


3. Given log N = 8.8080 - 10. To find N. 

The mantissa .8080 lies between two tabulated values, 8075 and 
8082, and hence we interpolate. 

The given mantissa is 5/7 of the 
way from the first to the second of 
these values in the Tables. The 10 
difference of the corresponding 
numbers 6420 and 6430 in the 

Tables is 10. Hence we add the correction x = 5/7 X 10 = 7 
to 6420 and get 6427. Since the characteristic is. -2, units’ place 
is two places to the left of the 6. Hence N = .06427. 

Instead of interpolating as we did, we could use the marginal 
table under Prop. Pts. on the right (p. 11). The difference 5 between 
the value 8075 in the Table and the given value 8080 is found in 
the row 64 in both columns 7 and 8 of this marginal table. Under 
the agreement to make the correction even when we have a choice, 
we take 8 as the fourth digit, and this is to be placed after the 
number 642 which corresponds to the mantissa 8075, giving 6428. 
Hence N = .06428. 




LOGARITHMS 


169 


The values of N found by the methods of the two preceding para¬ 
graphs differ by a unit in the last place. Hereafter we shall use 
the second method. 

4. Given log N = 9.58065 - 10, to find N. 

We look in the five-place Table for the mantissa .58065. We find 
on page 79 that it lies between two 


tabulated values, 58058 and 58070, 

N 

log N 

being 7/12 of the way from the 

r r 38070 

580581 " 

former to the latter. The desired 10 


58065 J 7 

number is 7/12 of the way from 

38080 

58070 


38070 to 38080; the correction is 
x —7/12 X 10 = 6, and thus we 

get 38076. Since the characteristic is —1, the decimal point pre¬ 
cedes the 3, and we have 

N = .38076. 

The interpolation could have been accomplished by use of the 
proportional parts (Prop. Pts.) table in the margin on page 79. The 
tabular difference is 58070 — 58058 = 12; the partial difference is 
58065 — 58058 = 7. In the Prop. Pts. column headed 12, we find a 
number as near 7 as possible; it is 7.2; this occurs in row 6, which 
gives the correction. The interpolation should be done mentally. 

93. Products and quotients found by use of logarithms. 

We are now ready to use the fundamental laws of logarithms 
(p. 159) in computations. To compute a product we find 
the logarithms of the factors, add them to get the logarithm 
of the product, then find in a table the number of which that 
is the logarithm. 

Examples. — 1. To find N = 3.728 X .006378 by use of 
four-place logarithms. 

log 3.728 = 0.5714 
log .006378 = 7.8047 - 10 
log N = 8.3761 - 10 
N = .02378. 





170 


TRIGONOMETRY 


To compute a quotient we use Law II (p. 159). We 
find the logarithms of the numerator and denominator, and 
subtract the latter from the former, getting the logarithm of 
the quotient. The number of which this is the logarithm is 
found in the Tables; it is the required quotient. 

49 79 

2 . To find N = b Y use of a four -P lace table of 
logarithms. 


The characteristic of log 42.73 is written as 11 — 10 so that the 
subtraction will be possible without use of a negative sign except 
with the —10. 


log 42.73 = 11.6307 - 10 

log 3697 = 3.5678 

log N = 8.0629 - 10 

N = .01156. 


3. 


To find x = 


.38275 X .048293 
.062191 X 8346.8 


by use of a five-place 


table of logarithms. 


Calling the numerator N and the denominator D, we carry out 
the computation as follows: 

log .38275 = 9.58?92 - 10 log .062191 = 8.79373 - 10 
log .048293 = 8.68389 - 10 log 8346.8 = 3.92152 

log N = 18.26681 - 20 log D = 12.71525 - 10 

log D = 12.71525 - 10 

log x = 5.55156 - 10 x = .000035609 


^94. Cologarithms. Division may be carried out in a 
slightly different way. Instead of subtracting the logarithm 
of the denominator, we may add the negative of that loga¬ 
rithm. When the latter is written so that the decimal part 
is positive it is called the cologarithm of the number. Thus 

colog TV = — logiY, 
and the law for division becomes 
M 

l°g = log M + colog N. 








LOGARITHMS 


171 


The following examples will show how the cologarithm is 
found. 

Examples. — 1. To find colog 376.4 to four places. 

We find log 376.4 = 2.5757. We get the cologarithm by adding 
the negative of this to 10.0000 — 10: 

10.0000 - 10 
- log 376.4 = -2.5757 
colog 376.4 = 7.4243 - 10 

2. To find colog .006259 to five places. 

10.00000 - 10 
- log .006259 = -7.79650 + 10 
colog .006259 = 2.20350 

It is seen that the cologarithm may be found by starting at 
the left of the logarithm and subtracting each digit from 9 
until we come to the last which is different from zero; this 
one is subtracted from 10 and the subsequent digits of the 
cologarithm are 0. Using this rule it is easy to write down 
the cologarithm directly from the Table, care being taken to 
include the characteristic. This work must be done mentally 
if cologarithms are to be used to advantage. 

Example 3 of the preceding section would be solved by use 
of cologarithms as follows: 

log .38275 - 9.58292 - 10 
log .048293 = 8.68389 - 10 
colog .062191 = 1.20627 
colog 8346.8 = 6.07848 - 10 
log x = 25.55156 - 30 
x = .000035609 

95. Powers and roots. The third law of logarithms 
(p. 159) enables us to find a power of a number. We take 
the logarithm of the number, multiply it by the exponent, 
getting the logarithm of the power, and find the number 
corresponding to that logarithm. 





172 


TRIGONOMETRY 


Example. — To find x = (.3728) 5 . 

Using a four-place table we have 

log .3728 = 9.5714 - 10• 
multiplying by 5 gives 

log x = 47.8570 - 50 
x = .007194. 

The student should also solve this problem by use of five-place 
tables and obtain 

* = .0072012. 


The fourth law of logarithms (p. 159) is used in extracting 
roots. To find the rth root of a number, take the logarithm 
of the number, divide it by r to obtain the logarithm of the 
rth root, and find the corresponding number. 

Example. — To find V^728; -^.3728. 


Using five-place tables we have 

log .3728 = 19.57148 - 20; 

dividing by 2 gives 

log V.3728 = 9.78574 - 10, 
V^728 = .61057. 


Also 

log .3728 = 29.57148 - 30; 

dividing by 3 gives 

log ^.3728 = 9.85716 - 10, 
\/J$728 = .71972. 


We wrote the negative characteristic in each problem in such a way 
that after the division the only negative part of the logarithm 
was —10. 


*96. Computations involving negative numbers. We 

have remarked that negative numbers do not have loga¬ 
rithms. To obtain a product or quotient involving negative 
numbers, we may find the numerical value by disregarding 





LOGARITHMS 


173 


the signs, then subsequently prefixing the proper sign to the 
result. If there was an even number of negative factors, 
the sign should be +, if an odd number it should be —. 


EXERCISES 

Find the numbers whose logarithms are: 

1. (a) 2.4150; (b) 0.6785; (c) 9.9562 - 10. 

2. (a) 1.9031; (b) 0.6866; (c) 8.8222 - 10. 

3. (a) 1.44091; (b) 3.83715; (c) 8.68024 - 10. 

4. (a) 2.63144; (b) 0.80441; (c) 9.76020 - 10. 


Interpolate to find the numbers whose logarithms are: 


5. (a) 3.7508; 

6. (a) 4.6520; 

7. (a) 4.76010; 

8. (a) 7.43701; 

9. (a) 5.95266; 

10. (a) 2.07100; 


(b) 7.6752 - 10. 
(b) 8.8278 - 10. 
(b) 8.45356 - 10. 
(b) 7.79010 - 10. 
(b) 7.23008 - 10. 
(b) 9.83672 - 10. 


Make use of a four-place table of logarithms to find the 
values of the following expressions to four significant figures: 


11 . 31.8 X 561. 

13. 820.4 X .06297. 


15. 


375 

1250* 


17. (1.035) 10 . 
19. V375.2. 


21 . 


V: 


3/.0246 X .3827 


571.2 X 67.89 


12 . 

14. 

16. 


729 X 2.45. 
6.233 X .8291. 
48.48 
6060 * 


18. (3.162) 3 . 
20. v 7 .02847. 


22 . 


v/ 


.008431 
(.2573) 3 ’ 


Make use of a five-place table of logarithms to find the values 
of the following expressions to five significant figures: 


23 . 48.279 X .36177. 
6371.8 


24. 828.37 X .62593. 
_ .0068123 


25. 


45216 


.082761 











174 


TRIGONOMETRY 


27. (3.3333) 3 . 


28. (2.7183) 5 . 


29. V47.635 X 823.49. 


V-24 X V / .729 
32 ' (-8.17) X (-2.25) 



-6187. X 23.46 2 
31 * 3847 X (-31.48) 3 


97. Logarithms of trigonometric functions. The calcu¬ 
lations of trigonometry may be shortened by use of loga¬ 
rithms. For this purpose Tables are given of the logarithms 
of the sine, cosine, tangent and cotangent* of angles from 
0 ° to 90°. In case angles outside this range are encountered 
we apply the formulas of §§47-53 (pp. 79-89) to express the 
functions in terms of angles within this range, and then use 
the Tables. 

Table IV (p. 12) gives four-place logarithms of the 
functions at intervals of 10'. For angles from 0° to 45°, 
which are found in the first column, we read the functions 
at the top of other columns; for angles from 45° to 90°, 
found in the last column, we read the functions at the 
bottom. The third column, which is headed d V gives the 
change in the logarithm of the sine (L Sin) for a change of 1' 
in the angle; this aids in interpolations. The fifth column, 
headed c d 1', shows the common difference of the logarithms 
of the tangent and the cotangent for a change of 1' in the 
angle. The next to last column gives the corresponding 
difference for the logarithm of the cosine. 

The characteristic which is printed in the Table must 
be decreased by 10, the —10 having been omitted for sim¬ 
plicity of printing. 

Examples. — 1. To find log sin 23° 52' to four places. 

* If one needs the logarithm of the secant or cosecant, he may recall 
that these functions are reciprocals of the cosine and sine respectively and 
hence use the relations 

log sec A = log 1/cos A = -log cos A = colog cos A, 
log esc A = log 1/sin A = —log sin A = colog sin A. 







LOGARITHMS 


175 


On page 15 of the Tables we go down the first column to 23° 50', 
across to the column headed L Sin and read 9.6065. Since the 
difference for 1' between angles 23° 50' and 24° 00' is 2.8, the cor¬ 
rection for 2' is 2 X 2.8 = 6 approximately. And since the L Sin 
increases when the angle increases we add the correction. Hence 

log sin 23° 52' = 9.6071 - 10. 

2. To find log tan 52° 18' to four places. 

On page 17 of the Tables we find 52° 10' in the last column; we go 
across to the column having L Tan at the bottom, and read 10.1098. 
The difference for 1' between 52° 10' and 52° 20' is 2.6. Hence 
the correction for 8' is 8 X 2.6 = 21 approximately. Since L Tan 
increases when the angle increases from 52° 10' to 52° 20', we add 
the correction. The final result is 

log tan 52° 18' = 10.1119 - 10 = 0.1119. 

3. To find log cos 71° 33' to four places. 

On page 14 we find 71° 30' in the last column. Going across to the 
column having L Cos at the bottom we read 9.5015. The difference 
for 1' is 3.8 and hence for 3' it is 3 X 3.8 = 11 approximately. 
Since L Cos decreases when the angle increases from 71° 30' to 
71° 40' we subtract the correction. The final result is 

log cos 71° 33' = 9.5004 - 10. 

4. To find the acute angle A, given 

log cot A = 8.9843 — 10. 

On page 13 in the column having L Cot at the bottom, we find 
8.9966 and 8.9836. Hence A lies between the corresponding angles 
84° 20' and 84° 30'. The difference in the logarithms is (disre¬ 
garding the decimal point) 9966 — 9843 = 123; since the difference 
for 1' is 13.0, the correction to the angle is 123/13.0 = 9'. Hence 

A = 84° 29'. 

Table VI is a five-place table of the logarithms of func¬ 
tions, with angles given at intervals of 1'. On each page the 
number of degrees in the angle is read at the top or bottom, 


176 


TRIGONOMETRY 


the number of minutes at the left or right; interpolation is 
necessary for parts of a minute. The angles 0° to 44° are 
found at the tops of the pages, 89° to 45° at the bottoms. 

5. To find log sin and log cot of the angle 23° 41' 37". 

On page 50, which has 23° printed at the top, we find 
log sin 23° 41' = 9.60388, log sin 23° 42' = 9.60417. 

The required log sin lies between these two, whose difference is 29 
(see third column), the decimal point in the values of log sin being 
disregarded for simplicity in carrying out the interpolation. Since 
T = 60", the correction for 37" is 37/60 of 29. This may be found 
by use of the Prop. Pts. (proportional parts) tables. In the column 
headed 29 we find the correction for 30" to be 14.5, and for 7" to 
be 3.4; thus the total correction is 14.5 + 3.4 = 18. Since log sin 
increases as the angle increases from 23° 41' to 23° 42' the correction 
is added. Thus we find 

log sin 23° 41' 37" = 9.60406 - 10. 

Similarly the correction for log cot is 17.0 + 4.0 = 21. Since 
log cot decreases the correction is subtracted, and we get 

log cot 23° 41' 37" = 10.35770 - 10. 

6. To find log tan and log cos of the angle 54° 57' 42". 

On page 62, which has 54° at the bottom, we enter the column 
having log tan at the bottom, go up to the row having 57 in the last 
column, and find 

log tan 54° 57' = 10.15397. 

To interpolate, we note that the difference of successive values of 
log tan is 27. The correction for 40" is 18.0; for 2" it is 1/10 of 
that for 20"; thus for 42" it is 18.0 + 0.9 = 19. Since log tan 
increases when the angle increases, this is added and we get 

log tan 54° 57'42" = 10.15416 - 10. 

Similarly the correction for log cos is 12.0 + 0.6 = 13; since 
log cos decreases, we have 

log cos 54° 57' 42" = 9.75900 - 10. 


LOGARITHMS 


177 


7. To find the acute angle A, given 

log cos A = 8.77990 — 10. 

On page 28, in the column having log cos at the bottom we find 
8.78152 and 8.77943 corresponding to angles 86° 32' and 86° 33'. 
Hence A lies between these angles. The difference 78152 - 77990 
= 162; the tabular difference in the third column is 209. Hence 
the correction to the angle 86° 32' is 162/209 of 60". In the Prop. 
Pts. tables on page 29 we find in the column headed 209 the cor¬ 
rection 139.3 in the 40" row; the difference 162 - 139.3 = 22.7 is 
nearly equal to the entry in the 7" row, being nearer to this than 
to 20.9. Hence by these tables the correction is about 47" and we 
have 

A = 86° 32' 47". 

*98. Angles near 0 ° or 90°. A glance at Table VI shows 
that for small angles, from 0° to 2° or further, the differences 
in log sin, log tan, and log cot are large. It follows that 
interpolation will not be very accurate. The same remark 
applies for angles from 90° to 88° or further, for log cos, 
log tan, and log cot. On the other hand the differences are 
so small for log cos when angles are near zero that when the 
function is given, the angle is not well determined. For 
example log cos A = 9.99997 - 10 for all angles from 0° 37' 
to 0°43'. On this account, when a small angle is to be 
found it is desirable to use a formula, if possible, which will 
give the sine, tangent, or cotangent of the angle. Similarly, 
to determine an angle near 90° we should avoid a formula 
which gives its sine, but use one giving its cosine or tangent. 

To increase the accuracy of interpolation for angles near 
0° or 90° we use the special Table Vb (p. 22). This 
gives the values of log sin for angles at intervals of 10" 
from 0° to 3°. For angles from 0° to 3° we can find the 
values of log cos and log tan from the formulas 

log cos A — 10 — C — 10, 
log tan A = log sin A + C, 


178 


TRIGONOMETRY 


where C is a correction which is given in the Table. This 
formula gives an error of at most 1 in the last figure of the 
mantissa. For an angle from 87° to 90° use the cofunction 
of the complementary angle. 

Examples. — 1. To find log tan 0° 37' 43 // by use of Table 
Vb 

We find 

log tan 0° 37' 40''* = 8.03970 - 10, 
log tan 0° 37' 50" = 8.04162 - 10. 

The difference for 10" is 192; the correction for 3" is 
3/10 X 192 = 57.6 = 58 approximately. 

Hence log tan 0° 37' 43" = 8.04028 - 10. 


2 . To find B, given log tan B = 2.26170. 


The angle is near 90°. Let A be its complement, A = 90° — B. 
Then 


Hence 


log cot A = 2.26170. 

log tan A = 10 — log cot A — 10 
= 7.73830 - 10 


From Table Vb, 

log tan 0° 18' 40" = 7.73480 - 10 
log tan 0° 18' 50" = 7.73866 - 10. 
By interpolation we find 

A = 0° 18' 49.07". 


Hence 

B = 89° 41' 10.93". 


Interpolation in Tables Vb or VI may be avoided and 
higher accuracy attained by use of Table Va. 

3. To find log tan 0° 37' 43" by means of Table Va. 

We have the formula log tan A = log A' + T where A' is the 
number of minutes in the angle; here A' = 37.717. Then, by 
Table VII, 


log A' = 1.57654 


LOGARITHMS 


179 


and by Table Va 


Hence 


T = 6.46374 - 10. 
log tan 0° 37' 43" = 8.04028 - 10. 


4. To find A if log tan A = 2.26170, by Table Va. 

The angle is near 90°. We are to use the formula log cot A = 
Ti + log A/, where A / = 90° — A expressed in minutes. We have 

log cot A = 7.73830 - 10. 


From Table VI, A = 89° 41' approximately, 
approximately. From Table Va 


Since 


Tx = 6.46373 - 10. 


we have 

From Table VII 


log Ax = log cot A — Tx 
log Ax' = 1.27457. 


Hence 


Ax' = 18.818'. 


Ax = 18'49.08", 

A = 90° — Ax = 89° 41' 10.92". 


Hence Ax = 19' 


EXERCISES 

Find the following logarithms to four places: 


1. (a) log sin 17° 30'; 

2. (a) log tan 18° 10'; 

3. (a) log tan 59° 50'; 

4. (a) log sin 78° 40'; 
6. (a) log sin 38° 57'; 
6. (a) log tan 44° 44'; 


(b) log cos 43° 40'. 
(b) log cot 38° 50'. 
(b) log cot 72° 40'. 
(b) log cos 69° 20'. 
(b) log cot 68° 28'. 
(b) log cos 61° 27'. 


Find the following logarithms to five places: 

7. (a) log sin 28° 57'; (b) log cot 78° 28'. 

8 . (a) log tan 34° 44'; (b) log cos 71° 27'. 

9. (a) log sin 18° 27'35"; (b) log cos 68° 49'51". 


180 


TRIGONOMETRY 


10 . 

11 . 

12 . 

13 . 


(a) log tan 75° 37'22"; (b) log cot 9° 46'57". 

(a) log sin 84° 13' 45"; . (b) log tan 6° 16'16". 

(a) log cos 85° 12' 18"; (b) log cot 4° 21' 35". 

log sec 67° 14' 21". 14 . log esc 18° 58' 13". 


Find the following logarithms (a) by use of Table Va and 
( b ) by use of Table Vb. 

* 16 . log sin 1° 2'28". * 16 . log tan 0° 4'37". 

* 17 . log cos 88° 48'13.2". * 18 . log cot 89° 28'17.4 . 

Find the acute angle A by use of four-place tables from each 
of the following equations: 

19 . (a) log sin A = 9.4359 — 10; 

(b) log cot A = 9.7958 — 10. 

20. (a) log tan A = 1.2460; 

(b) log cos A = 9.8107 — 10. 

21. (a) log tan A = 8.9330 — 10; 

(b) log cot A = 0.4917. 

22. (a) log sin A = 8.9960 — 10; 

(b) log cos A = 9.7392 — 10. 

Find the acute angle A by use of five-place tables from each 
of the following equations: 

23 . (a) log tan A = 9.94627 — 10; 

(b) log cos A = 9.81250 — 10. 

24 . (a) log sin A — 9.87670 — 10; 

(b) log cot A = 0.26360. 

25 (a) log sin A = 9.61761 — 10; 

(b) log cos A = 8.79602 - 10. 

26 . (a) log cot A = 0.23980; 

(b) log tan A = 1.15982. 


Find the acute angle A by use (a) of Table Va, and (b) of 
Table Vb, from each of the following equations: 

+27. log sin A = 8.56191 — 10. 

+2S. log tan A = 8.20202 — 10. 


LOGARITHMS 


181 


*29. log cos A = 7.87990 — 10. 

*30. log cot A = 7.71017 - 10. 

*31. log tan A = 2.80808. 

*32. log cot A = 3.10101. 

★ 99. Change of base of logarithms. In a note at the 
end of § 88, we remarked that bases of logarithms other than 
10 may be used. How can we find the logarithm of a num¬ 
ber N to a base b, if its logarithm to a base a is known? 
We may arrive at the answer as follows: 

By definition 

b lo % b N = N. 


Take the logarithm of each number to the base a, using the 
third law of logarithms, § 89, to simplify the left member. 
We find that 


Hence 


loga N • log a b = log a N. 


log b N = 


log a N 
log a b ’ 


which answers our question. 

If in this formula we substitute N = a, and observe that 
loga cl — 1 , we have 


log*, a = 


1 

loga b 


Hence the preceding formula is equivalent to 
log b N = loga N • log£ a. 


If we take a = 10, b = e, where e is the base of natural 
logarithms (p. 158) we have the most important special 
case, 


logeiV = 


logio N _ logio N 
logio e “ .43429 


2.3026 logio N. 






182 


TRIGONOMETRY 


EXERCISES 

Find the values of the following logarithms: 

1. Iog 2 8. 2. logs 1/27. 3. logs 5. 

4 . (a) log, 4.278; (b) log, 42.78. 

5. (a) log, 3.607; (b) log, 360.7. 

6. log, .07241. 7. log, .82461. 

★ 100. The logarithmic scale. The student is familiar 

with an algebraic scale on a straight line; he will recall 
attaching numbers to points on the line in such a way that 
distances from a fixed point A are proportional to those 
numbers (Fig. 88). 

— 3 —2 —1 012345678 

Fig. 88. Algebraic scale. 

If numbers are placed on a line so that the distances from 
a point A are proportional to the logarithms of the numbers, 
we have a logarithmic scale (Fig. 89). 

1 2 4 n 8 10 16 20 

- 1 - 1 —I- 1 1 -' *“ 

A N T 

Fig. 89. Logarithmic scale. 

Suppose that to the points N and T the numbers n and 
10 are attached; then 

log n _ AN 
logTo ~~ ~AT' 

If we take A T as the unit of length, we have, since log 10 = 1, 
log n = AN. 

If the distance from A to the point marked n is designated 
by An, we have: 





LOGARITHMS 


183 



A 1 

= log 1 

= 0; 

A2 

= log 2 

= .301; 

A3 

= log 3 

= .477; 

33 

= log 4 

= .602; 

36 

= log 6 

= .778; 

38 

= log 8 

= .903; 

3To 

= log 10 

= 1; 

A100 

= log 100 

= 2. 

The final 

zero of a number beyond 


10 is generally omitted in printing 
the scale. 

★ 101. The slide rule. This is an 
instrument devised to facilitate log¬ 
arithmic calculations in which not 
more than three-place accuracy is 
required.* It consists of two parts 
shaped like rulers, one of which slides 
in grooves in the other. On each a 
logarithmic scale is marked off (scale 
A and scale B, Fig. 90). Logarithms 
of numbers are added by sliding one 
rule along the other. Thus to add 
log 3 and log 2.5, place the point 
marked 1 on the B scale opposite the 
3 on the A scale; then the 2.5 on the 
B scale is opposite a point on the A 
scale whose distance from point 1 is 
log 3 + log 2.5. Since the last named 
point is 7.5, we have 
log 7.5 = log 3+log 2.5 = log 3X2.5, 


_</ * Accuracy to three significant figures is 

possible on a good slide rule. Special types 

_ _J of slide rules have been invented which give 

Fig. 90 greater accuracy. 


























































































































184 


TRIGONOMETRY 


hence 

3 X 2.5 = 7.5. 

We read for the same setting opposite 4.7 (B scale) the 
product 3 X 4.7 = 14.1 (.A scale). Other products are 
found similarly. For quotients the process is reversed. 

On the C scale the numbers are twice as far apart as on 
the B scale. It follows, since log n 2 = 2 log n, that the 
numbers on the scale B are the squares of opposite numbers 
on scale C, and those on C are square roots of corresponding 
ones on B. Scale D is related to A as scale C is to B. On 
the other side of the slide, for some slide rules, scales for 
log sin and log tan are found. By their use trigonometric 
calculations can be made. 

For a full description of slide rules with directions for 
their use see the manuals of instrument makers. 


CHAPTER IX 


SOLUTION OF TRIANGLES BY LOGARITHMS 

In Chapters II and III we discussed the solution of 
triangles. In those chapters calculations were made by 
elementary arithmetical methods; we are now ready to use 
logarithms. For right triangles we shall need no new for¬ 
mulas. For oblique triangles, however, we shall replace 
some of the formulas of Chapter III by others which are 
better adapted for logarithmic computations. 

102. Solution of right triangles. Two triangles will be 
solved as illustrations. In the first the data are given to 
four significant figures, and we therefore use four-place 
logarithms to get requisite accuracy as briefly as possible. 
In the second, five-place data require the use of five-place 
logarithms. 

It saves time and tends to greater accuracy in computa¬ 
tions to outline the solution completely before referring to the 
Tables or doing any computing. In our plan we should 
make provision for every number that is to be written, so 
that the later computation requires only the filling in of the 
outline. A complete outline includes the formulas, and a 
place for estimates obtained from a construction of a 
triangle. 

We shall adopt the notation and methods of § 30 (p. 41). 

Examples. — 1. Given C = 90°, A = 64° 13', b = 371.4. 
To find B, a, c. 

The formulas to be used are: 

B = 90° — A, a = b tan A, c = • 


185 



186 


TRIGONOMETRY 


For a check, we select one of the formulas 

cos B =~, b 2 = c 2 — a 2 = (c — a) (c + a). 


The second is the better check since the first would use log a and 
log c, and would not check the use of the Table in finding a and c. 
A complete outline for the solution is given below. The para¬ 
graph following contains explanations. 



B 


C = 


Data 
A = 


b = 


Construction and estimates 

a 

c = a = B = 
Formulas 

C B = 90° — A, a = b tan A, c 


Fig. 91. 1 cm. = 200. 


~ cos A * 

Check b 2 = c 2 - a 2 = (c - a) (c + a). 
Logarithmic formulas 


log a = log b + log tan A, 
log c = log b — log cos A. 
Check 2 log b = log (c - a) + log (c + a ). 

Computation 


(1) A = 

(3) 

log b = 

(2) B = 

(5) 

(—) log cos A = 

(9) c = 

- (7) 

log c = 


(4) 

log b = 


(6) 

(+) log tan A = 

(10) a = 

Check 

- (8) 

log a = 

(11) c — a — 

(13) 

log (c - a) = 

(12) c + o = 

(14) 

(+) log (c + a) = 


(15) 

log (c 2 — a 2 ) = 


(16) 

2 log b = 








SOLUTION OF TRIANGLES BY LOGARITHMS 187 


In this outline the numbers in parentheses would be omitted in 
actually preparing to solve a triangle. These numbers have been 
inserted to show the order in which the various steps could be taken 
in the computation if we wish to save time. The symbol ( —) 
placed ahead of log cos A is to indicate that the quantity is sub¬ 
tracted from the one above. The (+ ) signs in other places similarly 
indicate additions. The purpose of the arrows is to show that 
log c and log a are found before the numbers c and a which occur 
in the respective lines with them. 

For a computer who is familiar with the laws of logarithms the 
“Logarithmic formulas” are not needed, and we shall omit them 
in later examples. 

The details of the computation follow: 


A = 64° 13' 

log b = 12.5699 - 10 

B = 25° 47' 

(—) log cos A = 9.6384 - 10 

c = 854.0 

<— log c = 2.9315 

a = 769.0 

log b = 2.5699 
(H-)logtanA = 0.3160 
«- log a = 2.8859 

Check 

c - a = 85.0 

log (c — a) = 1.9294 

c A a = 1623.0 

(+) log (c+o)= 3.2103 


log (c 2 — a 2 ) = 
2 log b = 


5.1397 

5.1398 


The computed values should be checked with the estimates before 
the logarithmic check is applied; by 
this means large errors may be detected. 

2. Given b = .27946, c = .38072. 

To find A, B, a. 

Construction and Estimates 
A = 42°, B = 48°, a = .26. 

Formulas 



cosA = -, B =90° —A , 
Check b 2 = c 2 — a 2 = 


a = b tan A. 
(c — a) (c + a). 


b =.27946 

Fig. 92. 1 cm. = .1 







188 


TRIGONOMETRY 


b = .27946 
c = .38072 
A = 42° 46' 25" 
B = 47°13'35" 


a = .25854 
Check 

c — a ~ .12218 
c + a = .63926 


Computation 

log b = 9.44632 - 10 
(—) log c = 9.58060 - 10 
<— log cos A = 9.86572 — 10 
log b = 9.44632 - 10 
(+) log tan A = 9.96621 — 10 
4 - log a = 9.41253 - 10 

log (c — a) = 9.08700 — 10 
(+) log (c + a) = 9.80568 — 10 
log (c 2 - a 2 ) = 18.89268 - 20 
2 log b = 18.89264 - 20 


The check is rather poor; on going over the computation again no 
error is detected. 


EXERCISES 

Write down the complete outline of the logarithmic solution 
of the right triangles in which the following parts are given 
(assuming C = 90°): 

1. A and a. 2. B and b. 3. A and c. 

4. B and c. 5. a and b. 6. a and c. 

Solve the following triangles by use of four-place logarithms; 
in each case C = 90°: 

1. A = 64° 30', a = 4630. 8. B = 51° 10', b = .629. 

9 . A = 87° 51', c = .4169. 10. B = 18° 37', c = .08192. 

11. a = 8726, b = 3194. 12. a = 34.65, c = 46.53. 

Use five-place logarithms to solve the following triangles; 
in each case C — 90°: 

13. A = 13° 23', a = 58.27. 14. B = 76° 7', b = .07432. 

15. A = 62° 27' 50", c = 2185.7. 

16. B = 88° 27' 40", c = .75437. 

17. a = 67.534, b = 42.379. 18. a = .21356, c = .92473. 





SOLUTION OF TRIANGLES BY LOGARITHMS 189 


103. The law of tangents.* 
we proved the law of sines, 


In Chapter III, § 40 (p. 59), 


( 1 ) 


sin A 
sin B ’ 


where a and b are any two sides of a triangle and A and B 
are the opposite angles. From this we derive another 
formula useful in the solving of oblique triangles by loga¬ 
rithms. 

Subtracting 1 from each member of (1) we obtain 

a — b _ sin A — sin B 
b sin B 

Adding 1 similarly gives 

a + b _ sin A + sin B 
b sin B 

Dividing the former of these equations by the latter, we 
have 

a — b _ sin A — sin B 
a + b sin A + sin B 

Apply formulas (6) and (5) of § 70 (p. 122) ; 

a — b 2 cos \ (A + B) sin \ (A — B) 


Hence 

( 2 ) 


a + b 2 sin \ (A + B) cos \ (A — B) 

a — b _ tan \ (A — B) 
a -I - b tan § (A -f- B) 


This formula is known as the law of tangents. It may be 
stated thus: In any triangle the difference of any two sides 
is to their sum as the tangent of one-half the difference of the 
opposite angles is to the tangent of one-half their sum. 

* If Chapter III has been omitted, §§ 38-41 should be taken up at this 
point. 













190 


TRIGONOMETRY 


In case a < b it is simpler to write the formula 
b - a _ tan \ {B — A) 
b + a tan | (B + A) ’ 

and avoid negative quantities. If the sides are designated 
by a and c, formula (2) becomes 

//IN a — c tan \ ( A — C) 

(4) a + c “ tan £ (A + C) ‘ 

A similar formula could be written with the letters b and c. 
EXERCISES 

Prove the following identities, in which a, b, c are the sides 
and A, B, C the opposite angles of any triangle: 
a — b _ 2 sin | C sin \ (A — B ) . 

1 ' b ~ sin B 

b sin B 

2 = _-— . 

c 2 sin | C cos \ C 

a — b _ sin \ (A — B ) . 
c cos | C 

4 a + b _ cos \ (A — B) * 
c sin | C 

Note. The formulas of Ex. 3 and Ex. 4 are called Mollweide’s 
equations. They are sometimes used in place of the law of tan¬ 
gents in checking a solution of a triangle. 


104. Solving oblique triangles by logarithms. The solv¬ 
ing of triangles reduces to four cases: 

Case I. Given two angles and one side. 

Case II. Given two sides and the angle opposite one of 
them. 

Case III. Given two sides and the included angle. 

Case IV. Given three sides. 

The logarithmic solution of each of Cases I, II, and III 
may be carried out and the results checked by use of the 
following three formulas: 













SOLUTION OF TRIANGLES BY LOGARITHMS 191 


1. A + B + C = 180°. 

2. The law of sines, equation (1), § 103. 

3. The law of tangents, equation (2), § 103. 

Before solving Case IV by use of logarithms new formulas 
will be developed (§§ 108, 109). 

105. Case I. Given two angles and one side. In this 
case the third angle is found at once from the formula 

A + B + C = 180°. 


The unknown sides may then be found by using the law of 
sines twice. The law of tangents in a form involving the 
two computed sides gives a good check.* 

Example. —Given A = 37° 13', B = 61° 58', 
a = 3.467. To find C, b, c. 

Construction and Estimates 
C = 83°; b = 5.0; c = 5.6 
Formulas 

C = 180° - (A + B) 

T a sin B a sin C 



a=3.467 
Fig. 93. 1cm. =2. 


sin A ’ sin A 

Check Since c > b, we take the law of tangents in the form 
c — b _ tan \ {C — B) 
c + b tan \ {C + B) 


a = 3.467 
A = 37° 13' 
B = 61° 58' 
A + B = 99° 11' 
C = 80° 49' 
c = 5.660 


b = 5.060 


Computation 


log a = 10.5400 - 10 
(—) log sin A = 9.7816 - 10 
log a/sin A = 0.7584 
(+) log sin C = 9.9944 — 10 
log c = 0.7528 
log a/sin A = 0.7584 
(+) log sin B = 9.9458 - 10 
log b = 0.7042 


* Some writers prefer to use one of Mollweide’s equations for a check. 










192 


TRIGONOMETRY 


Check 

c — b — 0.600 
c + h = 10.720 

C — B = 18° 51' 

C + B = 142° 47' 
i (C - B) = 9° 25.5' 

§ (C + R) = 71° 23.5' 


log (c - h) = 9.7782 
(-)log(c + 6) = 1.0302 

L = log^| = 8.7480 - 10 

°C + 0 


R 


log tan l (C - B) = 9.2201 - 10 
(—) log tan |(C + 5) = 0.4728 

~ g - 8.7473 - 10 


L and 1? are the logarithms of the two members of the check formula, 
and should be equal. The check is rather poor, but on going over 
the work again we find no error. Since c — b is known to only 
three significant figures, we cannot expect results to check to more 
than three figures. When the triangle is solved by use of five-place 
Tables, the results are 


C = 80° 49', b = 5.0597, c = 5.6588. 


EXERCISES 

1. Check the solution in the preceding Example by use of 
Mollweide’s formula, 

c + b _ cos \ (C — B) ' 
a sin J A 

2. Give a complete outline of the solution of the oblique 
triangle when B, C, and b are given. 

Use four-place logarithms to solve the triangle and check 
your results, when the following are given: 

3. A = 82° 14', B = 31° 16', c = 147.1. 

4. A = 58° 57', C = 60° 46', c = 48.79. 

6. B = 66° 23', C = 19° 51', a = 2.146. 

6. B = 107° 42', C = 62° 2', b = .02876. 

Use five-place logarithms to solve the triangle and check your 
results when the following are given: 








SOLUTION OF TRIANGLES BY LOGARITHMS 193 


7. B = 33° 42' 5", C = 79° 35' 35", 

8. A = 21° 13' 15", 5 = 82° 28' 55", 

9. A = 42° 4' 45", C = 18° 51' 25", 

10. A = 31° 8'25", B = 114° 14'45", 


a = 9876.3. 
b = 47.218. 
b = .48107. 
c = .020707, 


106. Case II. Given two sides and an angle opposite one 
of them.* Suppose the given parts are A, a, and b. The 
angle B can be found by use of the law of sines, 



sin A 

sin B 


a 

= b ’ 

whence 

(i) 

sin B 

b sin A 

a 


It is to be recalled that the sine of an angle is never greater 
than 1; hence log sin B is at most 0, and in general has a 
negative characteristic. If formula (1) gives a value larger 
than 0 for log sin B, there can be no triangle having the given 
parts. If log sin B = 0, then B = 90°. 

If log sin B has a negative characteristic, we must re¬ 
member that the equation (1) is satisfied both by an acute 
angle B h found from the Tables, and by the supplement of 
this angle, that is, by B 2 = 180° — Bi. This follows from 
the equation 

sin B 2 = sin (180° — Bi) = sin B\. 

We thus face the possibility of having two triangles, which 
we may call triangles ABiCi and AB 2 C 2 (see Fig. 95, p. 195). 
We designate their unknown parts by B h C i, ci, and B 2 , C 2 , 
c 2 , respectively. 

The angle B having been found, we determine C from the 
equation 

(2) C = 180° - (A + B). 

* A geometrical discussion of this case is given in § 44 (p. 66). 





194 


TRIGONOMETRY 


In case we have two possible angles, B i and B 2 , we use this 
formula to determine the corresponding angles C i and C 2 : 

Ci = 180° — (A + BO, 

C 2 = 180° - (A + B 2 ). 

It may turn out at this step that A B 2 > 180°, making C 2 
negative; since the angles of a triangle must be positive, we 
conclude that there is no triangle AB 2 C 2 . Hence under 
these conditions only one triangle exists. But if A + B 2 < 
180°, we proceed with the solution of two triangles. 

When B and C are found, we get c from the law of sines, 

c _ sin C 
a sin A ’ 

whence 

a sin C 

® c= isrx* 

In case there are two triangles we have 

_ a s ^ n Ci _ asinCj. 

Cl ~ sinAi ’ ° 2 sinA 2 

The law of tangents may be employed to check the solu¬ 
tion (or solutions, in case there are two); the formula 

b — c tan \ (B — C) 

W b + c~ tan | (B + C) 

should be used, since it relates all three of the computed 
parts, B, C, and c. In case c > b, the letters b and c, as well 
as B and C, should be interchanged in formula (4) in order 
to avoid negative quantities. 

The student will find it helpful to construct a figure 
before outlining the computation, for by so doing he can 
usually tell in advance whether there will be no solution, 
one solution, or two solutions, and can draw up his plan 
accordingly. 








SOLUTION OF TRIANGLES BY LOGARITHMS 195 



Examples. — 1 . Given 
A = 47° 13', a = .2063, 
b = .7081. To find B, C, c. 

Construction and Estimates 
No solution. 

Formula 

. ^ b sin A 

sin B =- 


Computation 

log b = 9.8501 - 10 
(+) log sin A = 9.8657 - 10 
log b sin A = 9.7158 - 10 
(—) log a = 9.3145 - 10 
log sin B = 0.4013 

There is no angle B satisfying this equation, and hence no triangle 

having the given parts. 

2. Given A = 47° 13', 
a = .6063, b = .7081. To 
find B, C, c. 

Construction and Estimates 
Two solutions: 



sin B = 
Ci = 

Ci = 

Check 

ci — b _ 
Ci + b 



Bi = 60° 

B 2 = 120° 

. 1 cm. = .2. 

Ci = 73° 

C 2 = 13° 


ci = .81 

c 2 = .17 

Formulas 

b sin A 

a 


180“ - (A + Si), 

C 2 = 180° - (A + B 2 ) 

a sin Ci 

a sin C 2 


sin A ’ 

° 2 ~ sin A 


tan HCi — Bi) 

b — c 2 tan 

h (B 2 - C 2 ) 

tan \ (Ci + Bi) ’ 

b + c 2 tan 

\ {B 2 t C 2 ) 














196 


TRIGONOMETRY 


Computation 


log b = 

9.8501 - 10 

(+) log sin A = 

9.8657 - 10 

log b sin A = 

19.7158 - 20 

(-) log a = 

9.7827 - 10 

log sin B = 

9.9331 - 10 

B ! = 59° O' 

B 2 = 180° - Bx = 121° 0' 

A + Bi = 106° 13' A + Bi = 168° 13' 

Cl = 73° 47' 

C 2 = 11° 47' 

log sin Cx = 9.9824 — 10 

log sin C 2 = 9.3101 - 10 

(+) log a = 9.7827 - 10 

(+) log a = 9.7827 - 10 

log a sin Ci = 19.7651 — 20 

log a sin C 2 = 19.0928 — 20 

(—) log sin A = 9.8657 - 10 

(—) log sin A == 9.8657 - 10 

log ci = 9.8994 — 10 

log c 2 = 9.2271 - 10 

c, = .7932 

c 2 = .1687 

Check 

For brevity designate the left members of the check formulas by 

Li and L 2 , the right by Ri and R 2 


ci = .7932 

b = .7081 

b = .7081 

c 2 m .1687 

ci - b = .0851 

b - c 2 = .5394 

ci + b = 1.5013 

b + c 2 = .8768 

Ci - B x = 14° 47' 

B 2 - C 2 = 109° 13' 

Ci + Bx = 132° 47' 

B 2 + C 2 = 132° 47' 

h (Ci - BO = 7° 23.5' 

1 (B 2 - C 2 ) = 54° 36.5' 

| (Ci + Bi) = 66° 23.5' 

1 (B 2 + C 2 ) = 66° 23.5' 

log (ci - b) = 8.9299-10 

log (6 - c 2 ) = 19.7319 - 20 

log (ci + b) = 0.1765 

log ( b + c 2 ) = 9.9429 - 10 

log Li = 8.7534-10 

log L 2 = 9.7890 - 10 

log tan \ (Ci - B x ) = 9.1130-10 

log tan \ (Ci + Bx) = 0.3595 

log tan | (B 2 -C 2 ) = 10.1485 - 10 
log tan |(B 2 +C 2 ) = 0.3595 

log Rx = 8.7535-10 

log R 2 = 9.7890 - 10 

Since log Lx = log Rx nearly, and log L 2 = log R 2 , the solutions 


check. 














SOLUTION OF TRIANGLES BY LOGARITHMS 197 


3. Given A = 132° 47', a = 
angle C. 


Construction and Estimate 
One solution. C = 12°. 


sin B 


Formulas 
b sin A 


Ci = 180° - (A + B 0, 
C 2 = 180° - (A + B t ). 


To find sin A we use the relation 


,9063, b = .7081. To find the 



sin 132° 47' = sin (180° - 132° 47') = sin 47° 13'. 


Computation 
log b = 9.8501 - 10 
(+) log sin A = 9.8657 - 10 
log b sin A = 19.7158 - 20 
(—) log a = 9.9573 - 10 
log sin B = 9.7585 - 10 

Bi = 34° 59' B 2 = 180° - B l = 145° 1' 

A + Bi = 167° 46' A + B 2 = 277° 48' 

Ci = 12° 14' C 2 impossible 


, EXERCISES 

Solve the following triangles by use of four-place logarithms , 
given: 


1. 

A = 27° 10', 

a = 147.0, 

b = 468.0. 

2. 

C = 81° 5', 

a = 365.4, 

c = 317.2. 

3. 

B = 38° 19', 

a = 5617, 

b = 3863. 

4. 

A = 54° 12', 

a = 2.464, 

b = 4.027. 

5. 

B = 44° 9', 

b = .3818, 

c = .3025. 

e. 

C = 65° 12', 

a = 18.78, 

c = 19.38. 

7. 

A = 125° 11', 

a = 44.27, 

b = 55.87. 

8. 

B = 136° 10', 

b = 8471, 

c = 9462. 

9. 

C = 147° 12', 

a = 4.129, 

c = 5.681. 

10. 

B = 105° 5', 

a = .2076, 

b = .3592. 






198 


TRIGONOMETRY 


Solve the following triangles by use of five-place logarithms , 
given: 


11. 

A = 24° 15' 10", 

a = 12.474, 

b = 25.916. 

12. 

B = 78° 12' 45", 

6 = 367.29, 

c = 401.28. 

13. 

C = 42° 4' 15", 

a = 4.9761, 

c = 4.4226. 

14. 

A = 15° 8' 10", 

u = 289.87, 

c = 402.67. 

15. 

B = 43° 13' 55", 

a = .027472, 

b = .045825. 

16. 

C = 78° 12' 20", 

a = 248.27, 

c = 313.47. 

17. 

A = 157° 21' 40", 

a = .23654, 

6 = .48253. 

18. 

B = 110° 11' 30", 

b = 6.5219, 

c = 7.8261. 

19. 

C = 123° 4' 35", 

b = 234.25, 

c = 417.92. 

20. 

A = 161° 29' 5", 

a = 4.2734, 

b = 2.1494. 


107. Case III. Given two sides and the included angle. 

Suppose the given parts are a, b, C. By use of the formula 

A + B = 180° - C 

we find (A + B), then \ (A + B). The law of tangents, 
written in the form 

tan J (A — B) = tan I (A + B), 

is used to find § (A — B). By adding J (A - B) and 
§ (A + B) we get A; by subtracting \ (A — B) from 
i (A + B), we obtain B. The law of sines, in the form 

a sin C 
c = — — j -, 
smA 

enables us to compute c. We use 

A + B + C = 180° and b sin C = c sin B 
as check formulas. 

Example. — Given a = 77.99, b = 83.39, C = 72° 16\ 
To find c, A, B. 




SOLUTION OF TRIANGLES BY LOGARITHMS 199 


Construction and Estimates 
c = 93; A = 53°; B = 55°. 

Formulas 

A +B = 180° - C. 

Since b is greater than a we write the law 
of tangents 

tan h(B - A) = g^|tan I (B+ A), 

_ a sin C 
C ~ sin A 

Check A + B + C = 180°; b sin C = 



c sin B. 


Computation 


b 

= 

83.39 




a 

= 

77.99 




b — a 

= 

5.40 

log (b - a) = 

0.7324 


b + a 

= 

161.38 

(-)log(fe +a) = 

2.2078 


C 

= 

72° 16' 

. b — a 
log 6+a- 

8.5246 - 

10 

B + A 

= 

107° 44' 




1 (B + A) 

= 

53° 52' 

(+) log tan \ {B+A) = 

0.1366 


HB-A ) 

= 

2° 38' 

<— log tan l (.B —A) = 

8.6612 - 

10 

B 

= 

56° 30' 

log a = 

1.8920 


A 

= 

51° 14' 

(+) log sin C = 

9.9788 - 

10 

C 

= 

72° 16' 

log a sin C = 

11.8708 - 

10 

A +B +C 

= 

180° 00' 

( —) log sin A = 

9.8919 - 

10 

c 

= 

95.26 

«- log c = 

1.9789 





(+) log sin B = 

9.9211 - 

10 




R = log c sin B = 

1.9000 





log b = 

1.9211 





(+) log sin C = 

9.9788 - 

10 

L 

= 

R nearly. 

L = log b sin C = 

1.8999 



EXERCISES 


Solve and check each of the following triangles , using four- 
place logarithms: 
l.o = 74.80, 

2. b = 218.3, 


b = 66.30, C = 32° 57'. 

c = 127.5, A = 52° 13'. 















200 


TRIGONOMETRY 

3. 

a = 4571, 

c = 2818, 

B = 46° 46'. 

4. 

a = 2.185, 

b = 4.826, 

C = 12° 18'. 

5. 

b = .3174, 

c = .1247, 

A = 62° 16'. 

6. 

b = .04171, 

c = .5421, 

A = 132° 15'. 

7. 

a = 645.7, 

c = 124.8, 

B = 154° 47'. 

8. 

a = 88.49, 

b = 9.362, 

C = 5° 11'. 

Solve and check each of the following triangles, using 

place logarithms: 



9. 

a = 363.82, 

b = 459.18, 

C = 42° 15' 35". 

10. 

b = 89.725, 

c = 62.318, 

A = 57° 11'20". 

11. 

a = 5.7290, 

c = 8.4732, 

B = 68° 14' 15". 

12. 

a = .82497, 

b = .53261, 

C = 31° 18' 55". 

13. 

b = .071461, 

c = .099812, 

A = 12° 14' 15". 

14. 

a = 88.776, 

b = 14.82, 

C = 109° 18' 30". 

15. 

b = 462.31, 

c = 5481.2, 

A = 3° 13' 10". 

16. 

a = 38.876, 

c = .24172, 

B = 168° 14' 12". 


★ 108. The half-angle formulas. First proof. Before 
taking up the logarithmic solution of Case IV, in which the 

three sides a, b, c are given, 
we need to derive some new 
formulas. Draw the in¬ 
scribed circle in the tri¬ 
angle, Figure 98, calling its 
radius r. Then OA bisects 
the angle A, and we have 



( 1 ) 


tan l = if 


To express AF in terms of a, b, c, we note that the tan¬ 
gents from A are of equal length; hence AF = AE. Sim¬ 
ilarly BF = BD, CD = CE. Calling the perimeter of the 
triangle 2 s, we have 
2s = a-\~b-\-c 

= 2 AF + 2 BD + 2 CD = 2 AF + 2 (BD + CD), 
= 2 AF + 2 a. 


i 




SOLUTION OF TRIANGLES BY LOGARITHMS 201 


Hence 

and we have 

( 2 ) 


AF — s — a, 



To express r in terms of a, b, c , we proceed as follows. 
The area, S, of the triangle ABC is the sum of the areas of 
the triangles OAB, OBC, and OCA. Hence 


S = J rc + | ra + \ rb = \ r (a + b + c). 


Since a + 6 + c = 2s, we get 


(3) 


S = rs. 


From plane geometry we have the formula* 

S = Vs(s — a) (s — b) (s — c). 
Hence, from (3), 


(4) 


\/ 


(s — a) (s — b) (s — c) 


Formulas similar to (2) hold for the angles B and C. We 
thus have the three half-angle formulas 


(5) tan J A 


— r — , tan jB = - r , 
s — a s — b’ 


s — c 


where r is given by (4) and s = (a + b + c)/2. 

109. The half-angle formulas. Second proof. When the 
three sides, a, b , c, are given we may determine the angles 
by use of the law of cosines (§ 41, p. 60). Thus to find A , 
we have 

a 2 — b 2 + c 2 — 2 be cos A, 

* A proof of this formula is given in § 111. 









202 


TRIGONOMETRY 


whence 

( 1 ) 


eos A = 


b 2 + c 2 


2 be 


But this formula is not very well adapted to logarithmic 
calculation. A better formula is obtained as follows. 

From the formula (§ 69, p. 118) 


tan 


A-JL 

2 V 1 


— cos A 


+ cos A 


we find by substitution of the value of cos A given in (1) 
and by algebraic reduction, 


tan 


■Vi 

w- 


’2 be - b 2 ■ 

- c 2 + a 2 


2 be + b 2 ■ 

■f & - a 2 


fa 2 

- (b 2 - 

- 2 be + (?) 


(6 2 

+ 2 be 

+ c 2 ) - a 2 


\a 

- (6 - 

c)] [a + (b 

-C)] 

[(& 

+ c) - 

■ a] [(b + c) 

+ a] 

’{a 

-b + 

c) (a + b — 

c) 


(6 + c — a) (a + b + c) 

If we let s be the semi-perimeter of the triangle, then 

(2) 2 s = a + 6 + c, 2s — 2b = a — b + c, 

2s — 2a = b + c — a, 2s — 2c = a-\-b — c. 

Substituting these expressions in the preceding formula, 
we have 


tan 


A _ / 2 (s — b) 2 (s — c) 

2 V 2 (s — a) 2 s 




(s — a) (s — 6) (s 
(s — a) 2 s 


c) 


This may be written 


tan 


s — a 


(3) 


















SOLUTION OF TRIANGLES BY LOGARITHMS 203 


where 

(4) r = \J 
Similarly 

(5) 


(s - a) (s - b) (s - c) 

s 


s = 


a + b + c 


+ B 
tan 2 = 


r 

s^b’ 


tan 


s — c 


These are the half-angle formulas. 

110. Case IV. Given three sides. When the three sides 
are given, we first compute s and r from the relations 


(1) 2 s = a + b + c, r 2 — 


(s — a) (s — b) (s — c) 


and then find the angles A, B, C from the half-angle formula® 


( 2 ) tan 4 = 


r . B 
— , tan — = 


, C 
tan — = 


2 s — c 


2 s — a’ 2 s — b’ 
We check the results by the formula 
(3) A + B + C = 180°. 


We note that there will be no triangle if one given side is 
equal to or larger than the sum of the other two. When 
this impossible case arises, one of the factors in the numerator 
of the expression for r 2 is negative, and r is imaginary. 

Example. — 1. Given a = 513.4, b = 726.8, c = 931.3. 
To find A, B, C. ^ 

Construction and Estimates ~^ 

A = 34°, B = 49°, C = 97°. 

Formulas 

Equations (1), (2), (3). 



c=931.3 
Fig. 99. 1 cm. = 300. 












204 


TRIGONOMETRY 


s ■ 


- b 

— c 


a = 513.4 
b = 726.8 

c = 931.3 _ 

2 s = 2171.5 3s-2s = 
s = 1085.8 

log r = 2.2330 
(—) log (s — a) = 2.7577 
log tan \ A = 9.4753 
i A = 16° 38' 
log r = 2.2330 
(—) log (s — c) = 2.1889 
log tan \ C = 0.0441 
i C = 47° 54' 


Computation 

= 572.4 log (s - a) ■■ 
= 359.0 (+)log(s -b) ■ 
= 154.5 (+)log(s -c) 

= 1085.9 sum 

(-) logs 
log r 2 = 2 log r 
log r 

(-) log (s - b) 
— 10 log tan \ B 

iB 
A 
B 
C 

Check 180° 


= 2.7577 
= 2.5551 
: 2.1889 
: 7.5017 
= 3.0357 
= 4.4660 
= 2.2330 
= 2.5551 
= 9.6779 - 10 
= 25° 28' 

= 33° 16' 

= 50° 56' 

= 95° 48' 

= 180° 00' 


EXERCISES 

Solve the following triangles using four-place logarithms , 
or show that there will be no triangle: 


1. 

a = 72.4, 

b = 66.3, 

c = 81.9. 

2 . 

a = 3.08, 

b = 5.02, 

c = 4.27. 

3 . 

a = 8.256, 

b = 9.461, 

c = 9.109. 

4 . 

a = 6239, 

b = 7350, 

c = 8765. 

5 . 

a = .02457, 

b = .03176, 

c = .02887. 

6. 

a = 3.468, 

b = 2.816, 

c = 6.107. 

7 . 

a = 72.09, 

b = 35.02, 

c = 37.07. 

8 . 

a = 621.2, 

b = 187.5, 

c = 209.6. 


Solve the following triangles using five-place logarithms , 
or show that there will be no triangle: 


9 . 

a = 324.61, 

b = 421.72, 

c = 510.23. 

10 . 

a = 692.48, 

b = 536.11, 

c = 389.21. 

11 . 

a = 8.8762, 

b = 3.4271, 

c = 6.2471. 

12 . 

a = .97823, 

b = .86541, 

c = .21332. 

13 . 

a = 32.871, 

b = 42.107, 

c = 76.978. 

14 . 

a = 393.92, 

b = 292.93, 

c = 776.35. 










SOLUTION OF TRIANGLES BY LOGARITHMS 205 


111. Area of a triangle. Let S be the area of triangle 
ABC. Then, since (Fig. 100) 

he 


S 

we have 
( 1 ) 


2 ’ 


h = h sin A , 



This gives the area in terms of two sides and the included 
angle. 

The formula of plane geometry used without proof in 
§ 108 expresses S in terms of the three sides as follows: 


(2) S = Vs(s — a) (s — b) (s — c). 


This formula can be proved from relations established in 
: 108, 109. From § 109, equations (3) and (4), we have 


where r is given algebraically by the equation 

r-s/ 1 - 


\s — a) ( s — h) (s — c) 


In § 108 we showed that the former equation holds when r 
is interpreted as the radius of the inscribed circle. It fol¬ 
lows that the above formula for r gives this radius. In 
§ 108, however, formula (3) is S = rs. Hence 

S = s y/ ^ ~ ^ “ = Vs(s - a) (s - b) (s - c). 


To find the area of a triangle which falls under Case I 
or II we may first find an unknown side or angle and then 
apply formula (1). 

★ 112. Radii of inscribed and circumscribed circles. A 

formula for the radius r of the inscribed circle has been given 











206 


TRIGONOMETRY 


in § 111. From equations (5) and (4), § 108, we derive the 
following additional expressions: 

r = (s — a) tan \ A = (s — 6) tan \ B = (s — c) tan \ C. 


Let R be the radius of the circumscribed circle (Fig. 101), 
0 its center. Then by geometry ZBOC = 2 A, so that 
A = Z BOD, where OD is the perpendicu¬ 
lar bisector of BC. From the triangle BOD 
we therefore have 

. . a/2 . 

sin A = —> 

hence 



Similarly 


2 R 


a 

sin A 


2 R 


b 

sin B 1 


2 R 


c 

sin C 


Equating the expressions in the right members of the last 
three equations gives us the law of sines. 


EXERCISES 

1. By a method similar to that used in § 109 derive the 
formula 

. A _ 4 / (s - b) (s - c) 

Sin 2 “ V be 

2. In a similar manner prove that 

A 4 /s (s - a) 

C0S 2 V be 

3. From the formula sin A = 2 sin (A/2) cos (A/2), and 
formula (1), § 111, prove formula (2), § 111. Use the results 
of Exercises 1 and 2. 










SOLUTION OF TRIANGLES BY LOGARITHMS 207 


4. Prove that for any triangle 
e abc 

s “O' 


Find the areas 
'ven parts: 

of the triangles which have the following 

5. a = 10, 

c = 30, 

B = 25°. 

6. b = 20, 

c = 25, 

A = 55°. 

7. a = 75, 

b = 95, 

B = 105°. 

8. b = 128, 

c = 209, 

C = 48° 25'. 

9. A = 51°, 

B = 74°, 

a = 372. 

10. B = 76°, 

C = 42°, 

a = 208. 

11. a = 30, 

b = 40, 

c = 60. 

12. a = 212, 

b = 307, 

c = 188. 


113. Applications. In §37 (p. 52), we gave some appli¬ 
cations of right triangles, and at the end of Chapter III 
(p. 76) there are a number of miscellaneous exercises in¬ 
volving solutions of oblique triangles. The following set 
of exercises consists of further problems of these kinds, the 
first eight requiring the solution of right triangles, the others 
of oblique triangles. 


EXERCISES 

1. From a ship sailing a course of 55° (§ 6) at 8.2 mi. per 
hr., the bearing of a headland at 8:10 A.M. was due North, 
at 11:20 A.M. due West. How far was the ship from the 
headland at the latter hour? 

2. An army officer observes the angle of elevation of an 
airplane to be 62° 25', its distance to be 2125 yd. If a bomb 
drops vertically from the airplane, what is the horizontal 
distance from the officer to the point where it strikes? 

3. A surveyor measures the horizontal distance between 
two benchmarks as 486.32 ft. He finds one point to be 
27.375 ft. above the level of the other. Find the angle of 


208 


TRIGONOMETRY 


inclination of the line joining the two, and the distance be¬ 
tween them. 

4. Engineers propose to tunnel under a river, starting 
from a level 38.64 ft. above the bottom of a horizontal portion 
of the tunnel which is to be under the river, and giving an 
angle of inclination of the descent into the tunnel of 14° 30'. 
How long will one of the two sloping portions of the tunnel 



be? What is the horizontal distance from the beginning 
of the descent to the beginning of the horizontal portion of 
the tunnel? 

5. The radius of a circle is 32.52 mm. Find the angle at 
the center subtended by a chord of length 27.41 mm. 

6. Find the length of the circle of latitude that passes 
through Chicago, 41° 50' N, if the earth is a sphere of radius 
3959 mi. Also the length of the circle of latitude of Manila, 
14° 36' N. 

7. A man surveying a mine measures a line AB — 175 ft. 
from the mouth A of the mine due East at a dip of 14° 25' 
into the mine. From B he follows a tunnel BC 224 ft. along 
a line running due South at a dip of 25° 17'. How far is C 
below the level of A? If D is the point directly above C 
in the horizontal plane with A, what is the direction from 
A to D and how long is AD? 

8. A flagpole 25 ft. tall stands on the corner of a building 
132 ft. tall. Find the angle subtended by the flagpole from 
a point 325 ft. from the corner of the building in a horizontal 
line through its base. 






SOLUTION OF TRIANGLES BY LOGARITHMS 209 


9. A diagonal of a parallelogram is 15.24 in. long and 
at one of its ends it makes angles of 67° 41' and 44° 26' with 
the sides which meet there. Find the lengths of the 
sides. 

10. A tower 140.75 ft. high is situated on a hill. How far 
from the base of the tower is an object whose angles of de¬ 
pression from the top and the base of the tower are 29° 17' 30" 
and 21° 52' 45" respectively? 

11. From a boat an object A on the shore has the bear¬ 
ing S 41° 23' W. The boat goes due South at the rate of 
exactly 3 mi. per hr. At the end of 19 min. and 20 sec. 
the object at A has the bearing N 72° 45' W. How far was 
the boat from A at each observation? 

12. An observer at A notes that the angle of elevation of 
an airplane C due North of him is 43° 12' 25" at the same 
moment that an observer at B, 1125.3 ft. due South from A, 
finds that the elevation of C is 30° 27' 40". Find the dis¬ 
tances of C from A and B, and the height of C above the 
ground, assuming the line AS to be horizontal. 

13. Two points A and B on opposite shores of a lake are 
at known distances of 2.9661 mi. and 3.0426 mi. respectively 
from C. An observer at A finds that the angle BAC is 
64° 29' 35". Find the width of the lake from A to B. 

14. A triangle ABC is inscribed in a circle. The length of 
AB is 399.4 in., and that of BC is 415.2 in. The arc AB is 
exactly one-fifth of the whole circumference. Find the side 
AC and the angles A and B. 

15. The distance from A to a point C due West of A is 
not directly given, but is known to be about a quarter of a 
mile. Previous measurements from a point B have given 
BA = 7201.5 ft., BC = 6180.3 ft., and the bearing of B 
from A is N 48° 45' 35" W. Find AC. 

16. Astronomers knew that at a certain time the distance 
from the earth to the sun was 92,830,000 mi., and from the 
sun to Mars was 141,500,000. They observed that the 


210 


TRIGONOMETRY 


angle formed at the earth by lines toward the sun and Mars 
was 68° 29'. How far was Mars from the earth? 

17. Two sides of a parallelogram are 7.9235 ft. and 4.0312 ft. 
long respectively, and the angle between them is 79° 21/ 15". 
Find the lengths of the diagonals and the angles they make 
with the sides. 

18. To go from A’ s house to B’ s, A must walk 1675 ft. 
along one straight street, turn through an angle of 78° 39', 
and then walk 2056 ft. along another street. How much 
shorter would have been a straight line from start to finish? 

19. The hands of a clock are 3.250 ft. and 2.725 ft. long 
respectively. How far apart are their tips when the time 
is 2:35? 

20. A tight wire rope 57.324 ft. long reaches from the 
ground to a point on a pole. The height of the pole above 
the point where the rope is attached is 62.736 ft. The angle 
between pole and rope is 132° 15' 25". Find the angle of 
elevation of the top of the pole from the ground end of the 
rope. 

21. The point A is 5.296 mi. due North of B. and the 
distances from C to A and B respectively are 3.025 mi. 
and 4.917 mi. What is the bearing of C from B? 

22. Two buoys, A and B, on a lake are known to be 1210 
yd. apart, and one is due North of the other. An observer 
on a hill-top due North of the buoys observes with a range¬ 
finder that the distance to A is 3240 yd. and that the dis¬ 
tance to B is 4350 yd. What is the elevation of the ob¬ 
server above the lake, to the nearest ten yards?* 

23. A gas company proposes to build a cylindrical tank 
on a triangular piece of ground. The measurements of the 
piece which are most easily made are those of the sides. 
A surveyor finds that a = 78.369 ft., b = 82.198 ft., c = 
110.742 ft. What is the diameter of the tank of largest 
base which can be constructed on the ground? 

24. A boat sailed 372 yd. due East, then turned to the 


SOLUTION OF TRIANGLES BY LOGARITHMS 211 


left at an angle and sailed 571 yd., then turned to the left 
again and sailed back to the starting point a distance of 
418 yd. What was the bearing of each leg of the course? 

25. Two circles whose radii are 21.65 and 37.29 intersect, 
the angle between tangents at a point of intersection being 
18° 36'. Find the distance between their centers, and the 
length of their common chord. 

26. Two chords from a point A on a circle are of length 
37.26 and 82.19; the angle between them is 129° 13'. Find 
the radius of the circle. 

27. The angles of a triangle are 36°, 82° and 62°. The 
radius of the circumscribed circle is 25. Find the lengths 
of the sides. 

28. The perimeter of a triangle is 72, the radius of the 
inscribed circle is 12, and one angle is 47°. Find the lengths 
of the sides. 

Hint. Use a half-angle formula and Mollweide’s equations. 

29. Two angles of a triangle are 72° 14' and 66° 28'; 
the radius of the inscribed circle is 62.84 in. Find the 
lengths of the sides of the triangle. 

30. Two forces of 327.4 lb. and 632.8 lb. act at a point at 
an angle of 16° 37' with each other. What are the direction 
and the magnitude of the resultant force? 

31. Two forces of 36.2 lb. and 18.4 lb. act at a point A; 
they are exactly counteracted by a third force of 25.1 lb. 
Find the angles between the directions of the forces. 

32. A boat is traveling due East at the rate of 18 mi. 
per hr. A ball is thrown from the deck with a speed of 120 
ft. per sec. at an angle 37° to the right of the ship’s course. 
What is the speed and direction of the ball’s motion relative 
to the water? 

33. A man in an airplane which is traveling horizontally 
with a velocity of 165 mi. per hr. at an altitude of 8200 ft. 
throws a bomb directly downward at a rate of 200 ft. per sec. 


212 


TRIGONOMETRY 


Assuming that the velocity of the bomb is constant in mag¬ 
nitude and direction, how far will it strike from the point 
at which it was thrown? What is the angle of depression of 
its path? How long is it in the air? 

34. To find the height of a mountain top, P, above a 
horizontal plane ABC (Fig. 103), a base line AB was meas- 

p ured, AB = 3682 yd., and the angle of 
elevation of P from A observed to be 
21° 13'. The bearing of P from A was 
S 79° 18' E, and that of B from A was 
S 11° 34' E. The bearing of P from B 
was N 48° 16' E. What was the height 
of the mountain? 

35. Two sides of a triangle are 121.23 ft. and 197.56 ft. long 
respectively, and the angle between them is 121° 32' 15". 
Find the lengths of the segments into which the opposite 
side is cut by the bisector of the angle between the given 
sides. 



36. The frontage on the beach AB of a quadrangular lot 
ABCD cannot be measured directly. The sides BC, CD, 
DA are found to be 243 ft., 158 ft., Ill ft. respectively. 
The angles DAC and DBC are 33° 12' and 28° 40' respec¬ 
tively. Find the length of AB. 

37. A battleship starts from port A on a due easterly 
course at a speed of 18.2 mi. per hr. At the same instant a 
dispatch boat leaves port B at a speed of 24.3 mi. per hr. 
The bearing and distance of A from B are N 24° 10' E and 
37.2 mi. respectively. If the two boats continue at uniform 
speed, what should be the course of the dispatch boat so 
that it may meet the battleship? When will they meet? 

38. To find the height CP of a mountain top, P, above a 
horizontal plane ABC (Fig. 104), a line AD of length a was 
measured at an angle of inclination a with the horizontal, 
D being vertically above B. The angle of elevation of P 
from A was 9; angle CAB was /3, and angle ABC was y. 




SOLUTION OF TRIANGLES BY LOGARITHMS 213 


Show that 

pp _ a cos a sin 7 tan 9 
~ sin (0 + 7 ) 

39 . Devise a scheme for finding the distance between two 
accessible points A and B if there is no point from which 
both can be seen. (For example, A and B may lie on op¬ 
posite sides of an inaccessible mountain.) Assume that 
two points C and D can be found in a plane with A and B, 
such that A and D are visible from C, and B from D; also 
that AC, CD, DB can be measured. Give formulas to be 
used in finding A B from measured quantities. 




40 . Two astronomers in the same longitude observe the 
zenith distance of the center of the moon when it crosses 
their meridian. Their difference of latitude is 92° 14' 12"; 
the observed zenith distances are: A = 44° 54' 21 " and 
B = 48° 42' 57". Taking the earth's radius to be 3959 mi., 
find the distance from earth to moon (EM, Fig. 105). 






214 


TRIGONOMETRY 


FORMULAS 

Definitions of the six functions, p. 17. 

(1) sin 6 = V; ■ 

(3) tan0=|- 
Fig. 114 ( 5 ) sec 6 = ~' 



(2) cos 9 = 


(4) cot 9 = - 
v ' y 

(6) esc 9 = - 
y 


Reduction formulas, pp. 80-87. 

(7) sin (-0) = — sin 9, cos (-0) = cos0. 

(8) sin (90° - 0) = cos 0, cos (90° - 0) = sin 0. 

(9) sin (90° + 0) = cos0, cos (90° + 0) = -sin 0. 

(10) sin (180° - 0) = sin 0, cos (180° - 0) = - cos 


Formulas involving one angle, pp. 99-101. 



1 

cos 9 

1 

(11) 

sin 9 = - - 1 

esc 0 

sec0 


sin 0 

cot 9 

cos 9 

(12) 

tan 0 --- 1 

cos 9 

sin 9 

(13) 

sin 2 9 + cos 2 9 

= 1 . 


(14) 

1 + tan 2 9 = sec 2 6. 


(15) 

1 + cot 2 9 = esc 2 e. 



tan 0 = 


cot 0 


Addition formulas, p. 107. 


(16) 

(17) 

(18) 

(19) 

( 20 ) 
( 21 ) 


sin (a + j8) = sin a cos £ + cos a sin 0. 

sin (a — j8) = sin a: cos j8 — cos a sin |8. 

cos (a + j8) = cos a: cos /? — sin a sin /?. 

cos (a — (3) = cos a cos j8 + sin a sin 0. 

tan a + tan 0 

tan (a + 0) = 


tan (a — /3) = 


1 — tan a: tan j8 
tan a; — tan (3 
1 + tan a tan 0 


HI ^ 







FORMULAS 


215 


Formulas for the double angle, p. 116. 

( 22 ) sin 2 a = 2 sin a cos a. 

(23) cos 2 a = cos 2 a — sin 2 a 


tan 2 a = 


= 2 cos 2 a — 1 
= 1 — 2 sin 2 a. 
2 tan a: 


1 — tan 2 a 

Formulas for the half-angle, pp. 117, 118. 

(25) sin|=±y/- 

(26) cos|=±y/— 


COS a 


+ cos a 


(27) tan 


!- ± v't 


'1 — cos a 


2 V 1 + cos a 
1 — cos < 
sin a 
sin a 


1 + cos a 


Sums and differences expressed as products, p. 122c 

/ 00 \ • a . -r> • A —f - B A — B 

(28) sin A + sin B = 2 sin —-— cos —-- 

A A 

/ork\ • A • 7~> r* A -\- B . A — B 

(29) sm A — sin B = 2 cos —-— sm —-- 

z z 

a i d o A B A — B 

(30) cos A + cos B — 2 cos —-— cos 

z 

A 4 - B 

(31) cos A — cos B = —2 sin—-—sin 


2 

A — B 



Fig. 115 


2 2 
Radian measure, pp. 126-129. 

(32) 7 r radians = 180°. 

(33) s = rd (Fig. 115). 




















216 


TRIGONOMETRY 


Formulas for triangles. 


(34) 

(35) 

(36) 

(37) 

(38) 


, Law of sines, p. 60. 

1, Law of cosines, i 
, Law of tangents, p. 189. 


sin A sin B sin C 
a 2 = b 2 + & — 2 be cos A, Law of cosines, p. 61. 
a — b tan \ (A — B) 


a + b tan J ( A + B) 
tan^ = —-— , Half-angle formula, pp. 201, 202. 

A S d 

2s = a + b + c, ^ = (»-")(« — ft) (s -A 

s 

S = \ be sin A 


Vs(s — a ) (s — b) (s — c ), Area of triangle, p. 205. 










LOGARITHMIC 


AND 

TRIGONOMETRIC TABLES 

CONTENTS 


FOUR-PLACE TABLES 

PAGE 

Table I. Squares of Numbers ...... 2 

Table II. Values of Functions and Radians ... 4 
Table III. Logarithms of Numbers ..... 10 

Table IV. Logarithms of Functions.12 


FIVE-PLACE TABLES 

Logarithms of the Trigonometric Functions 


Table Va. Auxiliary Table of S and T . . . .21 

Table V5. Angles Near 0° and 90°.22 

Table VI. Logarithms of Functions .... 25 

Table VII. Common Logarithms of Numbers ... 73 

Table VIII. Natural Logarithms of Numbers ... 92 
Table IX. Constants with Their Logarithms ... 94 





2 


SQUARES OF NUMBERS 


I 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 

1.000 

1.020 

1.040 

1.061 

1.082 

1.103 

1.124 

1.145 

1.166 

1.188 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.8 
1.9 

1.210 

1.440 

1.690 

1.960 

2.250 

2.560 

2.890 

3.240 

3.610 

1.232 

1.464 

1.716 

1.988 

2.280 

2.592 

2.924 

3.276 

3.648 

1.254 

1.488 

1.742 

2.016 

2.310 

2.624 

2.958 

3.312 

3.686 

1.277 

1.513 

1.769 

2.045 

2.341 

2.657 

2.993 

3.349 

3.725 

1.300 

1.538 

1.796 

2.074 

2.372 

2.690 

3.028 

3.386 

3.764 

1.323 

1.563 

1.823 

2.103 

2.403 

2.723 

3.063 

3.423 

3.803 

1.346 

1.588 

1.850 

2.132 

2.434 

2.756 

3.098 

3.460 

3.842 

1.369 

1.613 

1.877 

2.161 

2.465 

2.789 

3.133 

3.497 

3.881 

1.392 

1.638 

1.904 

2.190 

2.496 

2.822 

3.168 

3.534 

3.920 

1.416 

1.664 

1.932 

2.220 

2.528 

2.856 

3.204 

3.572 

3.960 

2.0 

4.000 

4.040 

4.080 

4.121 

4.162 

4.203 

4.244 

4.285 

4.326 

4.368 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

2.8 
2.9 

4.410 

4.840 

5.290 

5.760 
6.250 

6.760 

7.290 

7.840 

8.410 

4.452 

4.884 

5.336 

5.808 

6.300 

6.812 

7.344 

7.896 

8.468 

4.494 

4.928 

5.382 

5.856 

6.350 

6.864 

7.398 

7.952 

8.526 

4.537 

4.973 

5.429 

5.905 

6.401 

6.917 

7.453 

8.009 

8.585 

4.580 

5.018 

5.476 

5.954 

6.452 

6.970 

7.508 

8.066 

8.644 

4.623 

5.063 

5.523 

6.003 

6.503 

7.023 

7.563 

8.123 

8.703 

4.666 

5.108 

5.570 

6.052 

6.554 

7.076 

7.618 

8.180 

8.762 

4.709 

5.153 

5.617 

6.101 

6.605 

7.129 

7.673 

8.237 

8.821 

4.752 

5.198 

5.664 

6.150 

6.656 

7.182 

7.728 

8.294 

8.880 

4.796 

5.244 

5.712 

6.200 

6.708 

7.236 

7.784 

8.352 

8.940 

3.0 

9.000 

9.060 

9.120 

9.181 

9.242 

9.303 

9.364 

9.425 

9.486 

9.548 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

3.9 

9.610 

10.24 
10.89 

11.56 

12.25 
12.96 

13.69 

14.44 

15.21 

9.672 

10.30 

10.96 

11.63 

12.32 

13.03 

13.76 

14.52 

15.29 

9.734 

10.37 
11.02 

11.70 

12.39 

13.10 

13.84 

14.59 

15.37 

9.797 

10.43 
11.09 

11.76 

12.46 

13.18 

13.91 

14.67 

15.44 

9.860 

10.50 

11.16 

11.83 

12.53 

13.25 

13.99 

14.75 

15.52 

9.923 

10.56 

11.22 

11.90 

12.60 

13.32 

14.06 

14.82 

15.60 

9.986 

10.63 

11.29 

11.97 

12.67 
13.40 

14.14 

14.90 

15.68 

10.05 

10.69 

11.36 

12.04 

12.74 

13.47 

14.21 

14.98 

15.76 

10.11 

10.76 

11.42 

12.11 

12.82 

13.54 

14.29 

15.05 

15.84 

10.18 

10.82 

11.49 

12.18 

12.89 

13.62 

14.36 

15.13 

15.92 

4.0 

16.00 

16.08 

16.16 

16.24 

16.32 

16.40 

16.48 

16.56 

16.65 

16.73 

4.1 

4.2 

4.3 

4.4 

4.5 

4.6 

4.7 

4.8 

4.9 

16.81 

17.64 

18.49 

19.36 

20.25 

21.16 

22.09 

23.04 

24.01 

16.89 

17.72 

18.58 

19.45 

20.34 

21.25 

22.18 

23.14 

24.11 

16.97 

17.81 

18.66 

19.54 

20.43 

21.34 

22.28 

23.23 

24.21 

17.06 

17.89 

18.75 

19.62 

20.52 

21.44 

22.37 

23.33 

24.30 

17.14 

17.98 

18.84 

19.71 

20.61 

21.53 

22.47 

23.43 

24.40 

17.22 

18.06 

18.92 

19.80 

20.70 

21.62 

22.56 

23.52 

24.50 

17.31 

18.15 

19.01 

19.89 

20.79 

21.72 

22.66 

23.62 

24.60 

17.39 

18.23 

19.10 

19.98 

20.88 

21.81 

22.75 

23.72 

24.70 

17.47 

18.32 

19.18 

20.07 

20.98 

21.90 

22.85 

23.81 

24.80 

17.56 

18.40 

19.27 

20.16 

21.07 

22.00 

22.94 

23.91 

24.90 

5.0 

25.00 

25.10 

25.20 

25.30 

25.40 

25.50 

25.60 

25.70 

25.81 

25.91 

5.1 

5.2 

5.3 

5.4 

26.01 

27.04 

28.09 

29.16 

26.11 

27.14 

28.20 

29.27 

26.21 

27.25 

28.30 

29.38 

26.32 

27.35 

28.41 

29.48 

26.42 

27.46 

28.52 

29.59 

26.52 

27.56 

28.62 

29.70 

26.63 

27.67 

28.73 

29.81 

26.73 

27.77 

28.84 

29.92 

26.83 

27.88 

28.94 

30.03 

26.94 

27.98 

29.05 

30.14 



































































































I 


SQUARES OF NUMBERS 


3 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.5 

5.6 

5.7 

5.8 

5.9 

30.25 

31.36 

32.49 

33.64 

34.81 

30.36 

31.47 

32.60 

33.76 

34.93 

30.47 

31.58 

32.72 

33.87 

35.05 

30.58 

31.70 

32.83 

33.99 

35.16 

30.69 

31.81 

32.95 

34.11 

35.28 

30.80 

31.92 

33.06 

34.22 

35.40 

30.91 

32.04 

33.18 

34.34. 

35.52 

31.02 

32.15 

33.29 

34.46 

35.64 

31.14 

32.26 

33.41 

34.57 

35.76 

31.25 

32.38 

33.52 

34.69 

35.88 

6.0 

36.00 

36.12 

36.24 

36.36 

36.48 

36.60 

36.72 

36.84 

36.97 

37.09 

6.1 

6.2 

6.3 

6.4 

6.5 

6.6 

6.7 

6.8 
6.9 

37.21 

38.44 

39.69 

40.96 

42.25 

43.56 

44.89 

46.24 

47.61 

37.33 

38.56 

39.82 

41.09 

42.38 
43.69 

45.02 

46.38 
47.75 

37.45 

38.69 

39.94 

41.22 

42.51 
43.82 

45.16 

46.51 
47.89 

37.58 

38.81 

40.07 

41.34 

42.64 
43.96 

45.29 

46.65 
48.02 

37.70 

38.94 

40.20 

41.47 

42.77 

44.09 

45.43 

46.79 

48.16 

37.82 

39.06 

40.32 

41.60 

42.90 

44.22 

45.56 

46.92 

48.30 

37.95 

39.19 

40.45 

41.73 

43.03 

44.36 

45.70 

47.06 

48.44 

38.07 

39.31 

40.58 

41.86 

43.16 

44.49 

45.83 

47.20 

48.58 

38.19 

39.44 

40.70 

41.99 

43.30 

44.62 

45.97 

47.33 

48.72 

38.32 

39.56 

40.83 

42.12 

43.43 

44.76 

46.10 

47.47 

48.86 

7.0 

49.00 

49.14 

49.28 

49.42 

49.56 

49.70 

49.84 

49.98 

50.13 

50.27 

7.1 

7.2 

7.3 

7.4 

7.5 

7.6 

7.7 

7.8 

7.9 

50.41 

51.84 

53.29 

54.76 
56.25 

57.76 

59.29 

60.84 

62.41 

50.55 

51.98 

53.44 

54.91 
56.40 

57.91 

59.44 
61.00 
62.57 

50.69 

52.13 

53.58 

55.06 

56.55 

58.06 

59.60 

61.15 

62.73 

50.84 

52.27 

53.73 

55.20 

56.70 

58.22 

59.75 

61.31 

62.88 

50.98 

52.42 

53.88 

55.35 

56.85 

58.37 

59.91 

61.47 

63.04 

51.12 

52.56 

54.02 

55.50 

57.00 

58.52 

60.06 

61.62 

63.20 

51.27 

52.71 

54.17 

55.65 

57.15 

58.68 

60.22 

61.78 

63.36 

51.41 

52.85 

54.32 

55.80 

57.30 

58.83 

60.37 

61.94 

63.52 

51.55 

53.00 

54.46 

55.95 

57.46 
58.98 

60.53 

62.09 

63.68 

51.70 

53.14 

54.61 

56.10 

57.61 

59.14 

60.68 

62.25 

63.84 

8.0 

64.00 

64.16 

64.32 

64.48 

64.64 

64.80 

64.96 

65.12 

65.29 

65.45 

8.1 

8.2 

8.3 

8.4 

8.5 

8.6 

8.7 

8.8 
8.9 

65.61 

67.24 
68.89 

70.56 

72.25 
73.96 

75.69 

77.44 

79.21 

65.77 

67.40 

69.06 

70.73 

72.42 

74.13 

75.86 

77.62 

79.39 

65.93 

67.57 
69.22 

70.90 

72.59 

74.30 

76.04 

77.79 

79.57 

66.10 

67.73 
69.39 

71.06 

72.76 

74.48 

76.21 

77.97 

79.74 

66.26 

67.90 

69.56 

71.23 

72.93 

74.65 

76.39 

78.15 

79.92 

66.42 

68.06 

69.72 

71.40 

73.10 
74.82 

76.56 

78.32 

80.10 

66.59 

68.23 

69.89 

71.57 

73.27 
75.00 

76.74 

78.50 

80.28 

66.75 

68.39 

70.06 

71.74 

73.44 

75.17 

76.91 

78.68 

80.46 

66.91 
68.56 
70.22 

71.91 
73.62 
75.34 

77.08 

78.85 

80.64 

67.08 

68.72 

70.39 

72.08 

73.79 

75.52 

77.26 

79.03 

80.82 

9.0 

81.00 

81.18 

81.36 

81.54 

81.72 

81.90 

82.08 

82.26 

82.45 

82.63 

9.1 

9.2 

9.3 

9.4 

9.5 

9.6 

9.7 

9.8 

9.9 

82.81 

84.64 

86.49 

88.36 

90.25 

92.16 

94.09 

96.04 

98.01 

82.99 

84.82 

86.68 

88.55 

90.44 

92.35 

94.28 

96.24 

98.21 

83.17 

85.01 

86.86 

88.74 

90.63 

92.54 

94.48 

96.43 

98.41 

83.36 

85.19 

87.05 

88.92 

90.82 

92.74 

94.67 

96.63 

98.60 

83.54 

85.38 

87.24 

89.11 

91.01 

92.93 

94.87 

96.83 

98.80 

83.72 

85.56 

87.42 

89.30 

91.20 

93.12 

95.06 

97.02 

99.00 

83.91 

85.75 

87.61 

89.49 

91.39 

93.32 

95.26 

97.22 

99.20 

84.09 

85.93 

87.80 

89.68 

91.58 

93.51 

95.45 

97.42 

99.40 

84.27 

86.12 

87.98 

89.87 

91.78 

93.70 

95.65 

97.61 

99.60 

84.46 

86.30 

88.17 

90.06 

91.97 

93.90 

95.84 

97.81 

99.80 





































































































4 FOUR-PLACE VALUES OF FUNCTIONS AND RADIANS II 


Degrees 

Radians 

Sin | 

Cos 

Tan | 

Cot 

Sec~[ 

Csc 



0° 00' 

.0000 

.0000 

1.0000 

.0000 


1.000 


1.5708 

90° 00' 

10 

029 

029 

000 

029 

343.8 

000 

343.8 

679 

50 

20 

058 

058 

000 

058 

171.9 

000 

171.9 

650 

40 

30 

.0087 

.0087 

1.0000 

.0087 

114.6 

1.000 

114.6 

1.5621 

30 

40 

116 

116 

.9999 

116 

85.94 

000 

85.95 

592 

20 

50 

145 

145 

999 

145 

68.75 

000 

68.76 

563 

10 

1° 00' 

.0175 

.0175 

.9998 

.0175 

57.29 

1.000 

57.30 

1.5533 

89° 00' 

10 

204 

204 

998 

204 

49.10 

000 

49.11 

504 

50 

20 

233 

233 

997 

233 

42.96 

000 

42.98 

475 

40 

30 

.0262 

.0262 

.9997 

.0262 

38.19 

1.000 

38.20 

1.5446 

30 

40 

291 

291 

996 

291 

34.37 

000 

34.38 

417 

20 

50 

320 

320 

995 

320 

31.24 

001 

31.26 

388 

10 

2° 00' 

.0349 

.0349 

.9994 

.0349 

28.64 

1.001 

28.65 

1.5359 

88° 00' 

10 

378 

378 

993 

378 

26.43 

001 

26.45 

330 

50 

20 

407 

407 

992 

407 

24.54 

001 

24.56 

301 

40 

30 

.0436 

.0436 

.9990 

.0437 

22.90 

1.001 

22.93 

1.5272 

30 

40 

465 

465 

989 

466 

21.47 

001 

21.49 

243 

20 

50 

495 

494 

988 

495 

20.21 

001 

20.23 

213 

10 

3° 00' 

.0524 

.0523 

.9986 

.0524 

19.08 

1.001 

19.11 

1.5184 

87° 00' 

10 

553 

552 

985 

553 

18.07 

002 

18.10 

155 

50 

20 

582 

581 

983 

582 

17.17 

002 

17.20 

126 

40 

30 

.0611 

.0610 

.9981 

.0612 

16.35 

1.002 

16.38 

1.5097 

30 

40 

640 

640 

980 

641 

15.60 

002 

15.64 

068 

20 

50 

669 

669 

978 

670 

14.92 

002 

14.96 

039 

10 

4° 00' 

.0698 

.0698 

.9976 

.0699 

14.30 

1.002 

14.34 

1.5010 

86° 00' 

10 

727 

727 

974 

729 

. 13.73 

003 

13.76 

981 

50 

20 

756 

756 

971 

758 

13.20 

003 

13.23 

952 

40 

30 

.0785 

.0785 

.9969 

.0787 

12.71 

1.003 

12.75 

1.4923 

30 

40 

814 

814 

967 

816 

12.25 

003 

12.29 

893 

20 

50 

844 

843 

964 

846 

11.83 

004 

11.87 

864 

10 

5° 00' 

.0873 

.0872 

.9962 

.0875 

11.43 

1.004 

11.47 

1.4835 

00 

w 

o 

o 

© 

10 

902 

901 

959 

904 

11.06 

004 

11.10 

806 

50 

20 

931 

929 

957 

934 

10.71 

004 

10.76 

777 

40 

30 

.0960 

.0958 

.9954 

.0963 

10.39 

1.005 

10.43 

1.4748 

30 

40 

989 

987 

951 

992 

10.08 

005 

10.13 

719 

20 

50 

.1018 

.1016 

948 

.1022 

9.788 

005 

9.839 

690 

10 

6° 00' 

.1047 

.1045 

.9945 

.1051 

9.514 

1.006 

9.567 

1.4661 

oo 

0 

O 

© 

10 

076 

074 

942 

080 

9.255 

006 

9.309 

632 

50 

20 

105 

103 

939 

110 

9.010 

006 

9.065 

603 

40 

30 

.1134 

.1132 

.9936 

.1139 

8.777 

1.006 

8.834 

1.4573 

30 

40 

164 

161 

932 

169 

8.556 

007 

8.614 

544 

20 

50 

193 

190 

929 

198 

8.345 

007 

8.405 

515 

10 

7° 00' 

.1222 

.1219 

.9925 

.1228 

8.144 

1.008 

8.206 

1.4486 

© 

o 

o 

CO 

oo 

10 

251 

248 

922 

257 

7.953 

008 

8.016 

457 

50 

20 

280 

276 

918 

287 

7.770 

008 

7.834 

428 

40 

30 

.1309 

.1305 

.9914 

.1317 

7.596 

1.009 

7.661 

1.4399 

30 

40 

338 

334 

911 

346 

7.429 

009 

7.496 

370 

20 

50 

367 

363 

907 

376 

7.269 

009 

7.337 

341 

10 

8° 00' 

.1396 

.1392 

.9903 

.1405 

7.115 

1.010 

7.185 

1.4312 

oo 

to 

o 

O 

© 

10 

425 

421 

899 

435 

6.968 

010 

7.040 

283 

50 

20 

454 

449 

894 

465 

6.827 

011 

6.900 

254 

40 

30 

.1484 

.1478 

.9890 

.1495 

6.691 

1.011 

6.765 

1.4224 

30 

40 

513 

507 

886 

524 

6.561 

012 

6.636 

195 

20 

50 

542 

536 

881 

554 

6.435 

012 

6.512 

166 

10 

9° 00' 

.1571 

.1564 

.9877 

.1584 

6.314 

1.012 

6.392 

1.4137 

o 

o 

o 

i-t 

00 



Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

Radians 

Degrees 


























II FOUR-PLACE VALUES OF FUNCTIONS AND RADIANS 5 


Degrees 

Radians 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 



9° 00' 

.1571 

.1564 

.9877 

.1584 

6.314 

1.012 

6.392 

1.4137 

81° 00' 

10 

600 

593 

872 

614 

197 

013 

277 

108 

50 

20 

629 

622 

868 

644 

084 

013 

166 

079 

40 

30 

.1658 

.1650 

.9863 

.1673 

5.976 

1.014 

6.059 

1.4050 

30 

40 

687 

679 

858 

703 

871 

014 

5.955 

1.4021 

20 

50 

716 

708 

853 

733 

769 

015 

855 

992 

10 

o 

o 

o 

O 

rl 

.1745 

.1736 

.9848 

.1763 

5.671 

1.015 

5.759 

1.3963 

80° 00' 

10 

774 

765 

843 

793 

576 

016 

665 

934 

50 

20 

804 

794 

838 

823 

485 

016 

575 

904 

40 

30 

.1833 

.1822 

.9833 

.1853 

5.396 

1.017 

5.487 

1.3875 

30 

40 

862 

851 

827 

883 

309 

018 

403 

846 

20 

50 

891 

880 

822 

914 

226 

018 

320 

817 

10 

11° 00' 

.1920 

.1908 

.9816 

.1944 

5.145 

1.019 

5.241 

1.3788 

79° 00' 

10 

949 

937 

811 

974 

066 

019 

164 

759 

50 

20 

978 

965 

805 

.2004 

4.989 

020 

089 

730 

40 

30 

.2007 

.1994 

.9799 

.2035 

4.915 

1.020 

5.016 

1.3701 

30 

40 

036 

.2022 

793 

065 

843 

021 

4.945 

672 

20 

50 

065 

051 

787 

095 

773 

022 

876 

643 

10 

12 00' 

.2094 

.2079 

.9781 

.2126 

4.705 

1.022 

4.810 

1.3614 

78° 00' 

10 

123 

108 

775 

156 

638 

023 

745 

584 

50 

20 

153 

136 

769 

186 

574 

024 

682 

555 

40 

30 

.2182 

.2164 

.9763 

.2217 

4.511 

1.024 

4.620 

1.3526 

30 

40 

211 

193 

757 

247 

449 

025 

560 

497 

20 

50 

240 

221 

750 

278 

390 

026 

502 

468 

10 

13° 00' 

.2269 

.2250 

.9744 

.2309 

4.331 

1.026 

4.445 

1.3439 

77° 00' 

10 

298 

278 

737 

339 

275 

027 

390 

410 

50 

20 

327 

306 

730 

370 

219 

028 

336 

381 

40 

30 

.2356 

.2334 

.9724 

.2401 

4.165 

1.028 

4.284 

1.3352 

30 

40 

385 

363 

717 

432 

113 

029 

232 

323 

20 

50 

414 

391 

710 

462 

061 

030 

182 

294 

10 

H 1 

o 

O 

© 

.2443 

.2419 

.9703 

.2493 

4.011 

1.031 

4.134 

1.3265 

76° 00' 

10 

473 

447 

696 

524 

3.962 

031 

086 

235 

50 

20 

502 

476 

689 

555 

914 

032 

039 

206 

40 

30 

.2531 

.2504 

.9681 

.2586 

3.867 

1.033 

3.994 

1.3177 

30 

40 

560 

532 

674 

617 

821 

034 

950 

148 

20 

50 

589 

560 

667 

648 

776 

034 

906 

119 

10 

15° 00' 

.2618 

.2588 

.9659 

.2679 

3.732 

1.035 

3.864 

1.3090 

75° 00' 

10 

647 

616 

652 

711 

689 

036 

822 

061 

50 

20 

676 

644 

644 

742 

647 

037 

782 

032 

40 

30 

.2705 

.2672 

.9636 

.2773 

3.606 

1.038 

3.742 

1.3003 

30 

40 

734 

700 

628 

805 

566 

039 

703 

974 

20 

50 

763 

728 

621 

836 

526 

039 

665 

945 

10 

16° 00' 

.2793 

.2756 

.9613 

.2867 

3.487 

1.040 

3.628 

1.2915 

74° 00' 

10 

822 

784 

605 

899 

450 

041 

592 

886 

50 

20 

851 

812 

596 

931 

412 

042 

556 

857 

40 

30 

.2880 

.2840 

.9588 

.2962 

3.376 

1.043 

3.521 

1.2828 

30 

40 

909 

868 

580 

994 

340 

044 

487 

799 

20 

50 

938 

896 

572 

.3026 

305 

045 

453 

770 

10 

17° 00 ' 

.2967 

.2924 

.9563 

.3057 

3.271 

1.046 

3.420 

1.2741 

73° 00' 

10 

996 

952 

555 

089 

237 

047 

388 

712 

50 

20 

.3025 

979 

546 

121 

204 

048 

357 

683 

40 

30 

.3054 

.3007 

.9537 

.3153 

3.172 

1.048 

3.326 

1.2654 

30 

40 

083 

035 

528 

185 

140 

049 

295 

625 

20 

50 

113 

062 

520 

217 

108 

050 

265 

595 

10 

18° 00' 

.3142 

.3090 

.9511 

.3249 

3.078 

1.051 

3.236 

1.2566 

72° 00' 



Cos 

Sin 

Cot 

Tan 

Csc 

| Sec 

Radians 

Degrees 

































6 FOUR-PLACE VALUES OF FUNCTIONS AND RADIANS II 


Degrees 

Radians 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 



18° 00' 

.3142 

.3090 

.9511 

.3249 

3.078 

1.051 

3.236 

1.2566 

72° 00' 

10 

171 

118 

502 

281 

047 

052 

207 

537 

50 

20 

200 

145 

492 

314 

018 

053 

179 

508 

40 

30 

.3229 

.3173 

.9483 

.3346 

2.989 

1.054 

3.152 

1.2479 

30 

40 

258 

201 

474 

378 

960 

056 

124 

450 

20 

50 

287 

228 

465 

411 

932 

057 

098 

421 

10 

19° 00' 

.3316 

.3256 

.9455 

.3443 

2.904 

1.058 

3.072 

1.2392 

71° 00' 

10 

345 

283 

446 

476 

877 

059 

046 

363 

50 

20 

374 

311 

436 

508 

850 

060 

021 

334 

40 

30 

.3403 

.3338 

.9426 

.3541 

2.824 

1.061 

2.996 

1.2305 

30 

40 

432 

365 

417 

574 

798 

062 

971 

275 

20 

50 

462 

393 

407 

607 

773 

063 

947 

246 

10 

20° 00' 

.3491 

.3420 

.9397 

.3640 

2.747 

1.064 

2.924 

1.2217 

70° 00' 

10 

520 

448 

387 

673 

723 

065 

901 

188 

50 

20 

549 

475 

377 

706 

699 

066 

878 

159 

40 

30 

.3578 

.3502 

.9367 

.3739 

2.675 

1.068 

2.855 

1.2130 

30 

40 

607 

529 

356 

772 

651 

069 

833 

101 

20 

50 

636 

557 

346 

805 

628 

070 

812 

072 

10 

21° 00' 

.3665 

.3584 

.9336 

.3839 

2.605 

1.071 

2.790 

1.2043 

69° 00' 

10 

694 

611 

325 

872 

583 

072 

769 

1.2014 

50 

20 

723 

638 

315 

906 

560 

074 

749 

985 

40 

30 

.3752 

.3665 

.9304 

.3939 

2.539 

1.075 

2.729 

1.1956 

30 

40 

782 

692 

293 

973 

517 

076 

709 

926 

20 

50 

811 

719 

283 

.4006 

496 

077 

689 

897 

10 

22° 00' 

.3840 

.3746 

.9272 

.4040 

2.475 

1.079 

2.669 

1.1868 

68° 00' 

10 

869 

773 

261 

074 

455 

080 

650 

839 

50 

20 

898 

800 

250 

108 

434 

081 

632 

810 

40 

30 

.3927 

.3827 

.9239 

.4142 

2.414 

1.082 

2.613 

1.1781 

30 

40 

956 

854 

228 

176 

394 

084 

595 

752 

20 

50 

985 

881 

216 

210 

375 

085 

577 

723 

10 

© 

© 

o 

co 

<N 

.4014 

.3907 

.9205 

.4245 

2.356 

1.086 

2.559 

1.1694 

67° 00' 

10 

043 

934 

194 

279 

337 

088 

542 

665 

50 

20 

072 

961 

182 

314 

318 

089 

525 

636 

40 

30 

.4102 

.3987 

.9171 

.4348 

2.300 

1.090 

2.508 

1.1606 

30 

40 

131 

.4014 

159 

383 

282 

092 

491 

577 

20 

50 

160 

041 

147 

417 

264 

093 

475 

548 

10 

24° 00' 

.4189 

.4067 

.9135 

.4452 

2.246 

1.095 

2.459 

1.1519 

66° 00' 

10 

218 

094 

124 

487 

229 

096 

443 

490 

50 

20 

247 

120 

112 

522 

211 

097 

427 

461 

40 

30 

.4276 

.4147 

.9100 

.4557 

2.194 

1.099 

2.411 

1.1432 

30 

40 

305 

173 

088 

592 

177 

100 

396 

403 

20 

50 

334 

200 

075 

628 

161 

102 

381 

374 

10 

25° 00' 

.4363 

.4226 

.9063 

.4663 

2.145 

1.103 

2.366 

1.1345 

65° 00' 

10 

392 

253 

051 

699 

128 

105 

352 

316 

50 

20 

422 

279 

038 

734 

112 

106 

337 

286 

40 

30 

.4451 

.4305 

.9026 

.4770 

2.097 

1.108 

2.323 

1.1257 

30 

40 

480 

331 

013 

806 

081 

109 

309 

228 

20 

50 

509 

358 

001 

841 

066 

111 

295 

199 

10 

26° 00' 

.4538 

.4384 

.8988 

.4877 

2.050 

1.113 

2.281 

1.1170 

64° 00' 

10 

567 

410 

975 

913 

035 

114 

268 

141 

50 

20 

596 

436 

962 

950 

020 

116 

254 

112 

40 

30 

.4625 

.4462 

.8949 

.4986 

2.006 

1.117 

2.241 

1.1083 

30 

40 

654 

488 

936 

.5022 

1.991 

119 

228 

054 

20 

50 

683 

514 

923 

059 

977 

121 

215 

1.1025 

10 

27° 00' 

.4712 

.4540 

.8910 

.5095 

1.963 

1.122 

2.203 

1.0996 

63° 00' 



Cos 

Sin 

Cot 

Tan 

Csc 

| Sec 

Radians 

Degrees 

































II FOUR-PLACE VALUES OF FUNCTIONS AND RADIANS 7 


Degrees 

Radians 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 



27° 00' 

.4712 

.4540 

.8910 

.5095 

1.963 

1.122 

2.203 

1.0996 

63° 00' 

10 

741 

566 

897 

132 

949 

124 

190 

966 

50 

20 

771 

592 

884 

169 

935 

126 

178 

937 

40 

30 

.4800 

.4617 

.8870 

.5206 

1.921 

1.127 

2.166 

1.0908 

30 

40 

829 

643 

857 

243 

907 

129 

154 

879 

20 

50 

858 

669 

843 

280 

894 

131 

142 

850 

10 

28° 00' 

.4887 

.4695 

.8829 

.5317 

1.881 

1.133 

2.130 

1.0821 

62° 00' 

10 

916 

720 

816 

354 

868 

134 

118 

792 

50 

20 

945 

746 

802 

392 

855 

136 

107 

763 

40 

30 

.4974 

.4772 

.8788 

.5430 

1.842 

1.138 

2.096 

1.0734 

30 

40 

.5003 

797 

774 

467 

829 

140 

085 

705 

20 

50 

032 

823 

760 

505 

816 

142 

074 

676 

10 

29° 00' 

.5061 

.4848 

.8746 

.5543 

1.804 

1.143 

2.063 

1.0647 

61° 00' 

10 

091 

874 

732 

581 

792 

145 

052 

617 

50 

20 

120 

899 

718 

619 

780 

147 

041 

588 

40 

30 

.5149 

.4924 

.8704 

.5658 

1.767 

1.149 

2.031 

1.0559 

30 

40 

178 

950 

689 

696 

756 

151 

020 

530 

20 

50 

207 

975 

675 

735 

744 

153 

010 

501 

10 

30° 00' 

.5236 

.5000 

.8660 

.5774 

1.732 

1.155 

2.000 

1.0472 

60° 00' 

10 

265 

025 

646 

812 

720 

157 

1.990 

443 

50 

20 

294 

050 

631 

851 

709 

159 

980 

414 

40 

30 

.5323 

.5075 

.8616 

.5890 

1.698 

1.161 

1.970 

1.0385 

30 

40 

352 

100 

601 

930 

686 

163 

961 

356 

20 

50 

381 

125 

587 

969 

.675 

165 

951 

327 

10 

31° 00' 

.5411 

.5150 

.8572 

.6009 

1.664 

1.167 

1.942 

1.0297 

59° 00' 

10 

440 

175 

557 

048 

653 

169 

932 

268 

50 

20 

469 

200 

542 

088 

643 

171 

923 

239 

40 

30 

—>5498— 

^5225 

.8526 

.6128 

1.632 

1.173 

1.914 

1.0210 

30 

40 

527 

250 

511 

168 

621 

175 

905 

181 

20 

50 

556 

275 

496 

208 

611 

177 

896 

152 

10 

32° 00' 

.5585 

.5299 

.8480 

.6249 

1.600 

1.179 

1.887 

1.0123 

58° 00' 

10 

614 

324 

465 

289 

590 

181 

878 

094 

50 

20 

643 

348 

450 

330 

580 

184 

870 

065 

40 

30 

.5672 

.5373 

.8434 

.6371 

1.570 

1.186 

1.861 

1.0036 

30 

40 

701 

398 

418 

412 

560 

188 

853 

1.0007 

20 

50 

730 

422 

403 

453 

550 

190 

844 

977 

10 

33° 00' 

.5760 

.5446 

.8387 

.6494 

1.540 

1.192 

1.836 

.9948 

57° 00' 

10 

789 

471 

371 

536 

530 

195 

828 

919 

50 

20 

818 

495 

355 

577 

520 

197 

820 

890 

40 

30 

.5847 

.5519 

.8339 

.6619 

1.511 

1.199 

1.812 

.9861 

30 

40 

876 

544 

323 

661 

501 

202 

804 

832 

20 

50 

905 

568 

307 

703 

1.492 

204 

796 

803 

10 

34° 00' 

.5934 

.5592 

.8290 

.6745 

1.483 

1.206 

1.788 

.9774 

56° 00' 

10 

963 

616 

274 

787 

473 

209 

781 

745 

50 

20 

992 

640 

258 

830 

464 

211 

773 

716 

40 

30 

.6021 

.5664 

.8241 

.6873 

1.455 

1.213 

1.766 

.9687 

30 

40 

050 

688 

225 

916 

446 

216 

758 

657 

20 

50 

080 

712 

208 

959 

437 

218 

751 

628 

10 

35° 00' 

.6109 

.5736 

.8192 

.7002 

1.428 

1.221 

1.743 

.9599 

55° 00' 

10 

138 

760 

175 

046 

419 

223 

736 

570 

50 

20 

167 

783 

158 

089 

411 

226 

729 

541 

40 

30 

.6196 

.5807 

.8141 

.7133 

1.402 

1.228 

1.722 

.9512 

30 

40 

225 

831 

124 

177 

.393 

231 

715 

483 

20 

50 

254 

854 

107 

221 

385 

233 

708 

454 

10 

36° 00' 

.6283 

.5878 

.8090 

.7265 

1.376 

1.236 

1.701 

.9425 

54° 00' 



Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

Radians 

Degrees 
















8 FOUR-PLACE VALUES OF FUNCTIONS AND RADIANS II 


Degkees 

Radians 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 



36° 00' 

.6283 

.5878 

.8090 

.7265 

1.376 

1.236 

1.701 

.9425 

54° 00' 

10 

312 

901 

073 

310 

368 

239 

695 

396 

50 

20 

341 

925 

056 

355 

360 

241 

688 

367 

40 

30 

.6370 

.5948 

.8039 

.7400 

1.351 

1.244 

1.681 

.9338 

30 

40 

400 

972 

021 

445 

343 

247 

675 

308 

20 

50 

429 

995 

004 

490 

335 

249 

668 

279 

10 

37° 00' 

.6458 

.6018 

.7986 

.7536 

1.327 

1.252 

1.662 

.9250 

53° 00' 

10 

487 

041 

969 

581 

319 

255 

655 

221 

50 

20 

516 

065 

951 

627 

311 

258 

649 

192 

40 

30 

.6545 

.6088 

.7934 

.7673 

1.303 

1.260 

1.643 

.9163 

30 

40 

574 

111 

916 

720 

295 

263 

636 

134 

20 

50 

603 

134 

898 

766 

288 

266 

630 

105 

10 

38° 00' 

.6632 

.6157 

.7880 

.7813 

1.280 

1.269 

1.624 

.9076 

52° 00' 

10 

661 

180 

862 

860 

272 

272 

618 

047 

50 

20 

690 

202 

844 

907 

265 

275 

612 

.9018 

40 

30 

.6720 

.6225 

.7826 

.7954 

1.257 

1.278 

1.606 

.8988 

30 

40 

749 

248 

808 

.8002 

250 

281 

601 

959 

20 

50 

778 

271 

790 

050 

242 

284 

595 

930 

10 

39° 00' 

.6807 

.6293 

.7771 

.8098 

1.235 

1.287 

1.589 

.8901 

51° 00' 

10 

836 

316 

753 

146 

228 

290 

583 

872 

50 

20 

865 

338 

735 

195 

220 

293 

578 

843 

40 

30 

.6894 

.6361 

.7716 

.8243 

1.213 

1.296 

1.572 

.8814 

30 

40 

923 

383 

698 

292 

206 

299 

567 

785 

20 

50 

952 

406 

679 

342 

199 

302 

561 

756 

10 

40° 00' 

.6981 

.6428 

.7660 

.8391 

1.192 

1.305 

1.556 

.8727 

50° 00' 

10 

.7010 

450 

642 

441 

185 

309 

550 

698 

50 ■ 

20 

039 

472 

623 

491 

178 

312 

545 

668 

40 

30 

.7069 

.6494 

.7604 

‘.8541 

1.171 

1.315 

1.540 

.8639 

30 

40 

098 

517 

585 

591 

164 

318 

535 

610 

20 

50 

127 

539 

566 

642 

157 

322 

529 

581 

10 

o 

© 

o 

T-l 

.7156 

.6561 

.7547 

.8693 

1.150 

1.325 

1.524 

.8552 

49° 00' 

10 

185 

583 

528 

744 

144 

328 

519 

523 

50 

20 

214 

604 

509 

796 

137 

332 

514 

494 

40 

30 

.7243 

.6626 

.7490 

.8847 

1.130 

1.335 

1.509 

.8465 

30 

40 

272 

648 

470 

899 

124 

339 

504 

436 

20 

50 

301 

670 

451 

952 

117 

342 

499 

407 

10 

42° 00' 

.7330 

.6691 

.7431 

.9004 

1.111 

1.346 

1.494 

.8378 

48° 00' 

10 

359 

713 

412 

057 

104 

349 

490 

348 

50 

20 

389 

734 

392 

110 

098 

353 

485 

319 

40 

30 

.7418 

.6756 

.7373 

.9163 

1.091 

1.356 

1.480 

.8290 

30 

40 

447 

777 

353 

217 

085 

360 

476 

261 

20 

50 

476 

799 

333 

271 

079 

364 

471 

232 

10 

43° 00' 

.7505 

.6820 

.7314 

.9325 

1.072 

1.367 

1.466 

.8203 

47° 00' 

10 

534 

841 

294 

380 

066 

371 

462 

174 

50 

20 

563 

862 

274 

435 

060 

375 

457 

145 

40 

30 

.7592 

.6884 

.7254 

.9490 

1.054 

1.379 

1.453 

.8116 

30 

40 

621 

905 

234 

545 

048 

382 

448 

087 

20 

50 

650 

926 

214 

601 

042 

386 

444 

058 

10 

44° 00' 

.7679 

.6947 

.7193 

.9657 

1.036 

1.390 

1.440 

.8029 

46° 00' 

10 

709 

967 

173 

713 

030 

394 

435 

999 

50 

20 

738 

988 

153 

770 

024 

398 

431 

970 

40 

30 

.7767 

.7009 

.7133 

.9827 

1.018 

1.402 

1.427 

.7941 

30 

40 

796 

030 

112 

884 

012 

406 

423 

912 

20 

50 

825 

050 

092 

942 

006 

410 

418 

883 

10 

45° 00' 

.7854 

.7071 

.7071 

1.000 

1.000 

1.414 

1.414 

.7854 

45° 00' 



Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

Radians 

Degrees 


































FOUR-PLACE LOGARITHMS 

OF 

NUMBERS 


10 


FOUR-PLACE LOGARITHMS OF NUMBERS 


III 














Proportional Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 8 9 

1.0 

0000 

043 

086 

128 

170 

212 

253 

294 

334 

374 

4 

8 12 

17 21 25 

goo 
.y o 

1 

414 

453 

492 

531 

569 

607 

645 

682 

719 

755 

4 

8 11 

15 19 23 

jf sH 

2 

792 

828 

864 

899 

934 

969 

*004 

*038 

*072 

*106 

3 

7 10 

14 17 21 

O 03 d 

ft b a> 

3 

1139 

173 

206 

239 

271 

303 

335 

367 

399 

430 

3 

6 10 

13 16 19 

h P m 
£ § * 

4 

461 

492 

523 

553 

584 

614 

644 

673 

703 

732 

3 

6 

9 

12 15 18 

a S-g 

1.5 

761 

790 

818 

847 

875 

903 

931 

959 

987 

*014 

3 

6 

8 

11 14 17 

m o 

6 

2041 

068 

095 

122 

148 

175 

201 

227 

253 

279 

3 

5 

8 

11 13 16 

•!h o cp © 

7 

304 

330 

355 

380 

405 

430 

455 

480 

504 

529 

2 

5 

7 

10 12 15 

^ 

<D C 

8 

553 

577 

601 

625 

648 

672 

695 

718 

742 

765 

2 

5 

7 

9 12 14 


9 

788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

2 

4 

7 

9 11 13 


2.0 

3010 

032 

054 

075 

096 

118 

139 

160 

181 

201 

2 

4 

6 

8 11 13 

15 17 19 

1 

222 

243 

263 

284 

304 

324 

345 

365 

385 

404 

2 

4 

6 

8 10 12 

14 16 18 

2 

424 

444 

464 

483 

502 

522 

541 

560 

579 

598 

2 

4 

6 

8 10 12 

14 16 17 

3 

617 

636 

655 

674 

692 

711 

729 

747 

766 

784 

2 

4 

6 

7 

9 11 

13 15 17 

4 

802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

2 

4 

5 

7 

9 11 

12 14 16 

2.5 

979 

997 

*014 

*031 

*048 

*065 

*082 

*099 

*116 

*133 

2 

4 

5 

7 

9 10 

12 14 16 

6 

4150 

166 

183 

200 

216 

232 

249 

265 

281 

298 

2 

3 

5 

7 

8 10 

11 13 15 

7 

314 

330 

346 

362 

378 

393 

409 

425 

440 

456 

2 

3 

5 

6 

8 

9 

11 12 14 

8 

472 

487 

502 

518 

533 

548 

564 

579 

594 

609 

2 

3 

5 

6 

8 

9 

11 12 14 

9 

624 

639 

654 

669 

683 

698 

713 

728 

742 

757 

1 

3 

4 

6 

7 

9 

10 12 13 

3.0 

771 

786 

800 

814 

829 

843 

857 

871 

886 

900 

1 

3 

4 

6 

7 

9 

10 11 13 

1 

914 

928 

942 

955 

969 

983 

997 

*011 

*024 

*038 

1 

3 

4 

5 

7 

8 

10 11 12 

2 

5051 

065 

079 

092 

105 

119 

132 

145 

159 

172 

1 

3 

4 

5 

7 

8 

9 11 12 

3 

185 

198 

211 

224 

237 

250 

263 

276 

289 

302 

1 

3 

4 

5 

7 

8 

9 11 12 

4 

315 

328 

340 

353 

366 

378 

391 

403 

416 

428 

1 

2 

4 

5 

6 

8 

9 10 11 

3.5 

441 

453 

465 

478 

490 

502 

514 

527 

539 

551 

1 

2 

4 

5 

6 

7 

9 10 11 

6 

563 

575 

587 

599 

611 

623 

635 

647 

658 

670 

1 

2 

4 

5 

6 

7 

8 10 11 

7 

682 

694 

705 

717 

729 

740 

752 

763 

775 

786 

1 

2 

4 

5 

6 

7 

8 9 11 

8 

798 

809 

821 

832 

843 

855 

866 

877 

888 

899 

1 

2 

3 

5 

6 

7 

8 9 10 

9 

911 

922 

933 

944 

955 

966 

977 

988 

999 

*010 

1 

2 

3 

4 

5 

7 

8 9 10 

4.0 

6021 

031 

042 

053 

064 

075 

085 

096 

107 

117 

1 

2 

3 

4 

5 

6 

8 9 10 

1 

128 

138 

149 

160 

170 

180 

191 

201 

212 

222 

1 

2 

3 

4 

5 

6 

7 8 9 

2 

232 

243 

253 

263 

274 

284 

294 

304 

314 

325 

1 

2 

3 

4 

5 

6 

7 8 9 

3 

335 

345 

355 

365 

375 

385 

395 

405 

415 

425 

1 

2 

3 

4 

5 

6 

7 8 9 

4 

435 

444 

454 

464 

474 

484 

493 

503 

513 

522 

1 

2 

3 

4 

5 

6 

7 8 9 

4.5 

532 

542 

551 

561 

571 

580 

590 

599 

609 

618 

1 

2 

3 

4 

5 

6 

7 8 9 

6 

628 

637 

646 

656 

665 

675 

684 

693 

702 

712 

1 

2 

3 

4 

5 

6 

7 7 8 

7 

721 

730 

739 

749 

758 

767 

776 

785 

794 

803 

1 

2 

3 

4 

5 

6 

7 7 8 

8 

812 

821 

830 

839 

848 

857 

866 

875 

884 

893 

1 

2 

3 

4 

5 

6 

7 7 8 

9 

902 

911 

920 

928 

937 

946 

955 

964 

972 

981 

1 

2 

3 

4 

4 

5 

6 7 8 

5.0 

990 

998 

*007 

*016 

*024 

*033 

*042 

*050 

*059 

*067 

1 

2 

3 

3 

4 

5 

6 7 8 

1 

7076 

084 

093 

101 

110 

118 

126 

135 

143 

152 

1 

2 

3 

3 

4 

5 

6 7 8 

2 

160 

168 

177 

185 

193 

202 

210 

218 

226 

235 

1 

2 

3 

3 

4 

5 

6 7 7 

3 

243 

251 

259 

267 

275 

284 

292 

300 

308 

316 

1 

2 

2 

3 

4 

5 

6 6 7 

4 

324 

332 

340 

348 

356 

364 

372 

380 

388 

396 

1 

2 

2 

3 

4 

5 

6 6 7 

|N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 8 9 








































































































































Ill 


FOUR-PLACE LOGARITHMS OF NUMBERS 


11 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional Parts 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.5 

7404 

412 

419 

427 

435 

443 

451 

459 

466 

474 

1 

2 

2 

3 

4 

5 

5 

6 

7 

6 

482 

490 

497 

505 

513 

520 

528 

536 

543 

551 

1 

2 

2 

3 

4 

5 

5 

6 

7 

7 

559 

566 

574 

582 

589 

597 

604 

612 

619 

627 

1 

1 

2 

3 

4 

5 

5 

6 

7 

8 

634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

1 

1 

2 

3 

4 

4 

5 

6 

7 

9 

709 

716 

723 

731 

738 

745 

752 

760 

767 

774 

1 

1 

2 

3 

4 

4 

5 

6 

7 

6.0 

782 

789 

796 

803 

810 

818 

825 

832 

839 

846 

1 

1 

2 

3 

4 

4 

5 

6 

6 

1 

853 

860 

868 

875 

882 

889 

896 

903 

910 

917 

1 

1 

2 

3 

3 

4 

5 

6 

6 

2 

924 

931 

938 

945 

952 

959 

966 

973 

980 

987 

1 

1 

2 

3 

3 

4 

5 

5 

6 

3 

993 

*000 

*007 

*014 

*021 

*028 

*035 

*041 

*048 

*055 

1 

1 

2 

3 

3 

4 

5 

6 

6 

4 

8062 

069 

075 

082 

089 

096 

102 

109 

116 

122 

1 

1 

2 

3 

3 

4 

5 

5 

6 

6.5 

129 

136 

142 

149 

156 

162 

169 

176 

182 

189 

1 

1 

2 

3 

3 

4 

5 

5 

6 

6 

195 

202 

209 

215 

222 

228 

235 

241 

248 

254 

1 

1 

2 

3 

3 

4 

5 

5 

6 

7 

261 

267 

274 

280 

287 

293 

299 

306 

312 

319 

1 

1 

2 

3 

3 

4 

5 

5 

6 

8 

325 

331 

338 

344 

351 

357 

363 

370 

376 

382 

1 

1 

2 

3 

3 

4 

4 

5 

6 

9 

388 

395 

401 

407 

414 

420 

426 

432 

439 

445 

1 

1 

2 

3 

3 

4 

4 

5 

6 

7.0 

451 

457 

463 

470 

476 

482 

488 

494 

500 

506 

1 

1 

2 

3 

3 

4 

4 

5 

6 

1 

513 

519 

525 

531 

537 

543 

549 

555 

561 

567 

1 

1 

2 

3 

3 

4 

4 

5 

6 

2 

573 

579 

585 

591 

597 

603 

609 

615 

621 

627 

1 

1 

2 

3 

3 

4 

4 

5 

6 

3 

633 

639 

645 

651 

657 

663 

669 

675 

681 

686 

1 

1 

2 

2 

3 

4 

4 

5 

5 

4 

692 

698 

704 

710 

716 

722 

727 

733 

739 

745 

1 

1 

2 

2 

3 

4 

4 

5 

5 

7.5 

751 

756 

762 

768 

774 

779 

785 

791 

797 

802 

1 

1 

2 

2 

3 

3 

4 

5 

5 

6 

808 

814 

820 

825 

831 

837 

842 

848 

854 

859 

1 

1 

2 

2 

3 

3 

4 

4 

5 

7 

865 

871 

876 

882 

887 

893 

899 

904 

910 

915 

1 

1 

2 

2 

3 

3 

4 

4 

5 

8 

921 

927 

932 

938 

943 

949 

954 

960 

965 

971 

1 

1 

2 

2 

3 

3 

4 

4 

5 

9 

976 

982 

987 

993 

998 

*004 

*009 

*015 

*020 

*025 

1 

1 

2 

2 

3 

3 

4 

4 

5 

8.0 

9031 

036 

042 

047 

053 

058 

063 

069 

074 

079 

1 

1 

2 

2 

3 

3 

4 

4 

5 

1 

085 

090 

096 

101 

106 

112 

117 

122 

128 

133 

1 

1 

2 

2 

3 

3 

4 

4 

5 

2 

138 

143 

149 

154 

159 

165 

170 

175 

ISO 

186 

1 

1 

2 

2 

3 

3 

4 

4 

5 

3 

191 

196 

201 

206 

212 

217 

222 

227 

232 

238 

1 

1 

2 

2 

3 

3 

4 

4 

5 

4 

243 

248 

253 

258 

263 

269 

274 

279 

284 

289 

1 

1 

2 

2 

3 

3 

4 

4 

5 

8.5 

294 

299 

304 

309 

315 

320 

325 

330 

335 

340 

1 

1 

2 

2 

3 

3 

4 

4 

5 

6 

345 

350 

355 

360 

365 

370 

375 

380 

385 

390 

1 

1 

2 

2 

3 

3 

4 

4 

5 

7 

395 

400 

405 

410 

415 

420 

425 

430 

435 

440 

1 

1 

2 

2 

3 

3 

4 

4 

5 

8 

445 

450 

455 

460 

465 

469 

474 

479 

484 

489 

0 

1 

1 

2 

2 

3 

3 

4 

4 

9 

494 

499 

504 

509 

513 

518 

523 

528 

533 

538 

0 

1 

1 

2 

2 

3 

3 

4 

4 

9.0 

542 

547 

552 

557 

562 

566 

571 

576 

581 

586 

0 

1 

1 

2 

2 

3 

3 

4 

4 

1 

590 

595 

600 

605 

609 

614 

619 

624 

628 

633 

0 

1 

1 

2 

2 

3 

3 

4 

4 

2 

638 

643 

647 

652 

657 

661 

666 

671 

675 

680 

0 

1 

1 

2 

2 

3 

3 

4 

4 

3 

685 

689 

694 

699 

703 

708 

713 

717 

722 

727 

0 

1 

1 

2 

2 

3 

3 

4 

4 

4 

731 

736 

741 

745 

750 

754 

759 

763 

768 

773 

0 

1 

1 

2 

2 

3 

3 

4 

4 

9.5 

777 

782 

786 

791 

795 

800 

805 

809 

814 

818 

0 

1 

1 

2 

2 

3 

3 

4 

4 

.I 

6 

823 

827 

832 

836 

841 

845 

850 

854 

859 

863 

0 

1 

1 

2 

2 

3 

3 

4 

4 

7 

868 

; 872 

877 

881 

886 

890 

894 

899 

903 

908 

0 

1 

1 

2 

2 

3 

3 

4 4 

8 

912 

: 917 

921 

926 

930 

934 

939 

943 

948 

952 

0 

1 

1 

2 

2 

3 

3 


rr 

9 

956 

i 961 

965 

969 

974 

978 

983 

987 

991 

996 

0 

1 

1 

2 

2 

3 

3 

3 

4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 



































































































































TABLE IV 


FOUR-PLACE LOGARITHMS 
OF TRIGONOMETRIC FUNCTIONS 

Note 1. — For simplicity in printing, all characteristics have been increased by 10. 
Hence 10 must be subtracted from each tabulated value of a logarithm. 

Note 2. — To avoid interpolating for angles between 0° and 3° or 87° and 90° 
use Tables V a or V 6. 


Angle 

L Sin 

dl' 

L Tan 

c d 1' 

L Cot 

L Cos 

dl' 


0° O' 

10' 

20' 

30' 

40' 

50' 

1° 0' 

10' 

20' 

30' 

40' 

50' 

2° 0' 

10' 

20' 

30' 

40' 

50' 

3° 0' 

10' 

20' 

30' 

40' 

50' 

4° 0' 

10' 

20' 

30' 

40' 

50' 

5° 0' 

7.4637 

.7648 

.9408 

8.0658 

.1627 

301.1 

176.0 

125.0 

96.9 

79.2 

66.9 

58.0 

51.1 

45.8 

41.3 

37.8 

34.8 

32.1 
30.0 
28.0 

26.3 

24.8 

23.5 

22.2 
21.2 
20.2 

19.2 

18.5 

17.7 

17.0 

16.3 

15.8 
15.2 
14.7 

7.4637 

.7648 

.9409 

8.0658 

.1627 

301.1 

176.1 
124.9 

96.9 

79.2 

67.0 

58.0 

51.2 

45.7 

41.5 

37.8 

34.8 

32.2 
30.0 
28.1 

26.3 

24.9 

23.5 

22.3 

21.3 
20.2 

19.4 

18.5 

17.8 

17.1 

16.5 

15.8 
15.4 

14.8 

12.5363 

.2352 

.0591 

11.9342 

.8373 

10.0000 

.0 

.0 

.0 

.0 

.0 

.1 

.0 

.0 

.0 

.1 

.0 

.1 

.0 

.1 

.0 

.1 

.0 

.1 

.1 

.0 

.1 

.1 

.1 

.1 

.0 

.1 

.1 

.1 

.1 

.2 

90° 0' 

50' 

40' 

30' 

20' 

10' 

89° 0' 

50' 

40' 

30' 

20' 

10' 

88° 0' 

50' 

40' 

30' 

20' 

10' 

87° 0' 

50' 

40' 

30' 

20' 

10 

86° 0' 

50' 

40' 

30' 

20' 

10' 

85° 0' 

.0000 

.0000 

.0000 

.0000 

.0000 

8.2419 

8.2419 

11.7581 

9.9999 

.3088 

.3668 

.4179 

.4637 

.5050 

.3089 

.3669 

.4181 

.4638 

.5053 

.6911 

.6331 

.5819 

.5362 

.4947 

.9999 

.9999 

.9999 

.9998 

.9998 

8.5428 

8.5431 

11.4569 

9.9997 

.5776 

.6097 

.6397 

.6677 

.6940 

.5779 
.6101 
.6401 
, .6682 
.6945 

.4221 

.3899 

.3599 

.3318 

.3055 

.9997 

.9996 

.9996 

.9995 

.9995 

8.7188 

8.7194 

11.2806 

9.9994 

.7423 

.7645 

.7857 

.8059 

.8251 

.7429 

.7652 

.7865 

.8067 

.8261 

.2571 

.2348 

.2135 

.1933 

.1739 

.9993 

.9993 

.9992 

.9991 

.9990 

8.8436 

8.8446 

11.1554 

9.9989 

.8613 

.8783 

.8946 

.9104 

.9256 

.8624 

.8795 

.8960 

.9118 

.9272 

.1376 

.1205 

.1040 

.0882 

.0728 

.9989 

.9988 

.9987 

.9986 

.9985 

8.9403 

8.9420 

11.058P 

9.9983 

1 

L Cos 

dl' 

L Cot 

c d 1' 

L Tan 

L Sin 

d 1' 

Angle 


12 






























































IV 


FOUR-PLACE LOGARITHMS OF FUNCTIONS 


13 


L Cot 

L Cos 

d 1' 


11.0580 

9.9983 

.1 

85° 0' 

.0437 

.9982 

50' 

.0299 

.9981 

.1 

40' 

.0164 

.9980 

.1 

30' 

.0034 

.9979 

.1 

20' 

10.9907 

.9977 

.2 

.1 

.1 

.2 

10' 

10.9784 

9.9976 

84° 0' 

.9664 

.9975 

50' 

.9547 

.9973 

40' 

.9433 

.9972 

.1 

30' 

.9322 

.9971 

. 1 

20' 

.9214 

.9969 

.2 

.1 

.2 

.2 

10' 

10.9109 

9.9968 

00 

CO 

o 

© 

.9005 

.9966 , 

50' 

.8904 

.9964 

40' 

.8806 

.9963 

.1 

30' 

.8709 

.9961 

.2 

20' 

.8615 

.9959 

.2 

.1 

.2 

.2 

10' 

10.8522 

9.9958 

3 

o 

O 

.8431 

.9956 

50' 

.8342 

.9954 

40' 

.8255 

.9952 

.2 

30' 

.8169 

.9950 

.2 

20' 

.8085 

.9948 

.2 

.2 

.2 

.2 

10' 

10.8003 

9.9946 

81° 0' 

.7922 

.9944 

50' 

.7842 

.9942 

40' 

.7764 

.9940 

.2 

30' 

.7687 

.9938 

.2 

20' 

.7611 

.9936 

.2 

.2 

.3 

.2 

10' 

10.7537 

9.9934 

© 

o 

O 

oo 

.7464 

.9931 

50' 

.7391 

.9929 

40' 

.7320 

.9927 

.2 

30' 

.7250 

.9924 

.3 

20' 

.7181 

.9922 

.2 

.3 

.2 

10' 

10.7113 

9.9919 

79° 0' 

.7047 

.9917 

.3 

50' 

.6980 

.9914 

40' 

.6915 

.9912 

.2 

30' 

.6851 

.9909 

.3 

20' 

.6788 

.9907 

.2 

.3 

.3 

.2 

10' 

10.6725 

9.9904 

78° 0' 

.6664 

.9901 

50' 

.6603 

.9899 

40' 

.6542 

.9896 

.3 

30' 

.6483 

.9893 

.3 

20' 

.6424 

.9890 

.3 

.3 

10' 

10.6366 

9.9887 

o 

© 

L Tan 

L Sin 

d 1' 

Angle 


Angle L Sin 


8 C 


O' 

10 ' 
20 ' 
30 ' 
40 ' 
50 ' 

0 ' 

10 ' 
20 ' 
30 ' 
40 ' 
50 ' 
0 ' 

10 ' 
20 ' 
30 ' 
40 ' 
50 ' 

1 0 ' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

9° 0' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

10 ° 0 ' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

11 ° 0 ' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

12 ° 0 ' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

13° O' 


8.9403 


.9545 

.9682 

.9816 

.9945 

9.0070 

9.0192 

.0311 

.0426 

.0539 

.0648 

.0755 

9.0859 

.0961 

.1060 

.1157 

.1252 

.1345 

9.1436 

.1525 

.1612 

.1697 

.1781 

.1863 

9.1943 

.2022 

.2100 

.2176 

.2251 

.2324 

9.2397 

.2468 

.2538 

.2606 

.2674 

.2740 

9.2806 

.2870 

.2934 

.2997 

.3058 

.3119 

9.3179 

.3238 

.3296 

.3353 

.3410 

.3466 


d 1' 


9.3521 


L Cos 


14.2 

13.7 

13.4 

12.9 

12.5 

12.2 

11.9 

11.5 

11.3 

10.9 

10.7 

10.4 

10.2 

9.9 

9.7 

9.5 

9.3 

9.1 

8.9 

8.7 

8.5 

8.4 

8.2 
8.0 

7.9 

7.8 

7.6 

7.5 
7.3 

7.3 

7.1 

7.0 

6.8 
6.8 

6.6 
6.6 

6.4 

6.4 
6.3 

6.1 
6.1 
6.0 

5.9 

5.8 

5.7 

5.7 

5.6 

5.5 


L Tan c d 1 


8.9420 


.9563 

.9701 

.9836 

.9966 

9.0093 

9.0216 

.0336 

.0453 

.0567 

.0678 

.0786 

9.0891 

.0995 

.1096 

.1194 

.1291 

.1385 

9.1478 

.1569 

.1658 

.1745 

.1831 

.1915 

9.1997 

.2078 

.2158 

.2236 

.2313 

.2389 

9.2463 

.2536 

.2609 

.2680 

.2750 

.2819 

9.2887 

.2953 
.3020 
• .3085 
.3149 
.3212 

9.3275 

.3336 

.3397 

.3458 

.3517 

.3576 


d 1' 


9.3634 


L Cot 


14.3 

13.8 

13.5 

13.0 

12.7 

12.3 

12.0 

11.7 

11.4 
11.1 

10.8 

10.5 

10.4 

10.1 

9.8 

9.7 
9.4 

9.3 

9.1 

8.9 

8.7 

8.6 

8.4 

8.2 

8.1 

8.0 

7.8 

7.7 

7.6 

7.4 

7.3 

7.3 

7.1 
7.0 

6.9 

6.8 

6.6 

6.7 

6.5 

6.4 
6.3 
6.3 

6.1 

6.1 

6.1 

5.9 
5.9 

5.8 


cd 1' 









































































14 


FOUR-PLACE LOGARITHMS OF FUNCTIONS 


IV 


Angle 

L Sin 

dl' 

L Tan 

c d 1' 

L Cot 

L Cos 

d 1' 


13° O' 

9.3521 

5.4 

9.3634 

5.7 

10.6366 

9.9887 

.3 

.3 

.3 

.3 

.3 

.3 

.3 

.3 

A 

77° 0' 

10' 

.3575 

.3691 

.6309 

.9884 

50' 

20' 

.3629 

5.4 

.3748 

5.7 

.6252 

.9881 

40' 

30' • 

.3682 

5.3 

.3804 

5.6 

.6196 

.9878 

30' 

40' 

.3734 

5.2 

.3859 

5.5 

.6141 

.9875 

20' 

50' 

.3786 

5.2 

5.1 

5.0 

.3914 

5.5 

5.4 

5.3 

.6086 

.9872 

10' 

14° 0' 

9.3837 

9.3968 

10.6032 

9.9869 

76° 0' 

10' 

.3887 

.4021 

.5979 

.9866 

50' 

20' 

.3937 

5.0 

.4074 

5.3 

.5926 

.9863 

40' 

30' 

.3986 

4.9 

.4127 

5.3 

.5873 

.9859 

.4 

.3 

o 

30' 

40' 

.4035 

4.9 

.4178 

5.1 

.5822 

.9856 

20' 

50' 

.4083 

4.8 

4.7 

4.7 

.4230 

5.2 

5.1 

5.0 

.5770 

.9853 

.3 

.4 

.3 

.3 

.4 

O 

10' 

o 

o 

iO 

iH 

9.4130 

9.4281 

10.5719 

9.9849 

75° 0' 

10' 

.4177 

.4331 

.5669 

.9846 

50' 

20' 

.4223 

4.6 

.4381 

5.0 

.5619 

.9843 

40' 

30' 

.4269 

4.6 

.4430 

4.9 

.5570 

.9839 

30' 

40' 

.4314 

4.5 

.4479 

4.9 

.5521 

.9836 

.3 

A 

20' 

50' 

.4359 

4.5 

4.4 

4.4 

.4527 

4.8 

4.8 

4.7 

.5473 

.9832 

.4 

.4 

.3 

.4 

A 

10' 

16° 0' 

9.4403 

9.4575 

10.5425 

9.9828 

74° 0' 

10' 

.4447 

.4622 

.5378 

.9825 

50' 

20' 

.4491 

4.4 

.4669 

4.7 

.5331 

.9821 

40' 

30' 

.4533 

4.2 

.4716 

4.7 

.5284 

.9817 

.4 

30' 

40 

.4576 

4.3 

.4762 

4.6 

.5238 

.9814 

.3 

A 

20' 

50' 

.4618 

4.2 

4.1 

4.1 

.4808 

4.6 

4.5 

4.5 

.5192 

.9810 

.4 

.4 

.4 

.4 

10' 

o 

o 

tH 

9.4659 

9.4853 

10.5147 

9.9806 

73° 0' 

10' 

.4700 

.4898 

.5102 

.9802 

50' 

20' 

.4741 

4.1 

.4943 

4.5 

.5057 

.9798 

40' 

30' 

.4781 

4.0 

.4987 

4.4 

.5013 

.9794 

.4 

30' 

40' 

.4821 

4.0 

.5031 

4.4 

.4969 

.9790 

.4 

A 

20' 

50' 

.4861 

4.0 

3.9 

3.9 

.5075 

4.4 

4.3 

4.3 

.4925 

.9786 

.4 

.4 

.4 

.4 

10' 

18° 0' 

9.4900 

9.5118 

10.4882 

9.9782 

72° 0' 

10' 

.4939 

.5161 

.4839 

.9778 

50' 

20' 

.4977 

3.8 

.5203 

4.2 

.4797 

.9774 

40' 

30' 

.5015 

3.8 

.5245 

4.2 

.4755 

.9770 

.4 

30' 

40' 

.5052 

3.7 

.5287 

4.2 

.4713 

.9765 

.5 

A 

20' 

50' 

.5090 

3.8 

3.6 

3.7 

.5329 

4.2 

4.1 

4.1 

.4671 

.9761 

.4 

.4 

.5 

.4 

10' 

19° 0' 

9.5126 

9.5370 

10.4630 

9.9757 

71° 0' 

10' 

.5163 

.5411 

.4589 

.9752 

50' 

20' 

.5199 

3.6 

.5451 

4.0 

.4549 

.9748 

40' 

30' 

.5235 

3.6 

.5491 

4.0 

.4509 

.9743 

.5 

30' 

40' 

.5270 

3.5 

.5531 

4.0 

.4469 

.9739 

.4 

20' 

50' 

.5306 

3.6 

3.5 

3.4 

.5571 

4.0 

4.0 

3.9 

.4429 

.9734 

.5 

.4 

.5 

.4 

10' 

o 

o 

O 

<N 

9.5341 

9.5611 

10.4389 

9.9730 

70° 0' 

10' 

.5375 

.5650 

.4350 

.9725 

50' 

20' 

.5409 

3.4 

.5689 

3.9 

.4311 

.9721 

40' 

30' 

.5443 

3.4 

.5727 

3.8 

.4273 

.9716 

.5 

30' 

40' 

.5477 

3.4 

.5766 

3.9 

.4234 

.9711 

.5 

20' 

50' 

.5510 

3.3 

3.3 

.5804 

3.8 

3.8 

.4196 

.9706 

.5 

.4 

10' 

21° 0' 

9.5543 

9.5842 

10.4158 

9.9702 

69° 0' 


L Cos 

d 1' 

L Cot 

c d r 

L Tan 

L Sin 

dl' 

Angle 



























































































V 


FOUR-PLACE LOGARITHMS OF FUNCTIONS 


15 


Angle 

L Sin 

21° 0' 

9.5543 

10' 

.5576 

20' 

.5609 

30' 

.5641 

40' 

.5673 

50' 

.5704 

22° 0' 

9.5736 

10' 

.5767 

20' 

.5798 

30' 

.5828 

40' 

.5859 

50' 

.5889 

23° 0' 

9.5919 

10' 

.5948 

20' 

.5978 

30' 

.6007 

40' 

.6036 

50' 

.6065 

24° 0' 

9.6093 

10' 

.6121 

20' 

.6149 

30' 

.6177 

40' 

.6205 

50' 

.6232 

to 

CJ1 

o 

© 

9.6259 

10' 

.6286 

20' 

.6313 

30' 

.6340 

40' 

.6366 

50' 

.6392 

26° 0' 

9.6418 

10' 

.6444 

20' 

.6470 

30' 

.6495 

40' 

.6521 

50' 

.6546 

27° 0' 

9.6570 

10' 

.6595 

20' 

.6620 

30' 

.6644 

40' 

.6668 

50' 

.6692 

© 

o 

oo 

<N 

9.6716 

10' 

.6740 

20' 

.6763 

30' 

.6787 

40' 

.6810 

50' 

.6833 

29° 0' 

9.6856 


L Cos 


d V 


3.3 

3.3 
3.2 
3.2 

3.1 

3.2 

3.1 

3.1 

3.0 

3.1 

3.0 

3.0 

2.9 

3.0 

2.9 

2.9 

2.9 

2.8 

2.8 

2.8 

2.8 

2.8 

2.7 

2.7 

2.7 

2.7 

2.7 

2.6 

2.6 

2.6 

2.6 

2.6 

2.5 

2.6 

2.5 

2.4 

2.5 

2.5 

2.4 

2.4 

2.4 

2.4 

2.4 

2.3 

2.4 
2.3 
2.3 
2.3 


d 1' 


L Tan 


9.5842 


.5879 

.5917 

.5954 

.5991 

.6028 


9.6064 


.6100 

.6136 

.6172 

.6208 

.6243 


9.6279 


.6314 

.6348 

.6383 

.6417 

.6452 


9.6486 

.6520 

.6553 

.6587 

.6620 

.6654 


9.6687 


.6720 

.6752 

.6785 

.6817 

.6850 


9.6882 


.6914 

.6946 

.6977 

.7009 

.7040 


9.7072 


.7103 

.7134 

.7165 

.7196 

.7226 


9.7257 


.7287 

.7317 

.7348 

.7378 

.7408 


9.7438 


L Cot 


cdr 


3.7 

3.8 
3.7 
3.7 
3.7 
3.6 

3.6 

3.6 

3.6 

3.6 

3.5 

3.6 

3.5 

3.4 

3.5 

3.4 

3.5 
3.4 

3.4 

3.3 

3.4 

3.3 

3.4 
3.3 
3.3 

3.2 

3.3 

3.2 

3.3 
3.2 

3.2 

3.2 

3.1 

3.2 

3.1 

3.2 

3.1 

3.1 

3.1 

3.1 

3.0 

3.1 

3.0 

3.0 

3.1 

3.0 

3.0 

3.0 


c d 1' 


L Cot 

L Cos 

d 1' 


10.4158 

9.9702 


69° O' 

.4121 

.9697 


50' 

.4083 

.9692 

.5 

40' 

.4046 

.9687 

.5 

30' 

.4009 

.9682 

.5 

20' 

.3972 

.9677 

.5 

.5 

10' 




10.3936 

9.9672 

.5 

© 

o 

co 

© 

.3900 

.9667 


50' 

.3864 

.9661 

.6 

40' 

.3828 

.9656 

.5 

30' 

.3792 

.9651 

.5 

20' 

.3757 

.9646 

.5 

.6 

10' 

10.3721 

9.9640 

.5 

67° 0' 

.3686 

.9635 


50' 

.3652 

.9629 

.6 

40' 

.3617 

.9624 

.5 

30' 

.3583 

.9618 

.6 

20' 

.3548 

.9613 

.5 

.6 

10' 

10.3514 

9.9607 

.5 

66° 0' 

.3480 

.9602 


50' 

.3447 

.9596 

.6 

40' 

.3413 

.9590 

.6 

30' 

.3380 

.9584 

.6 

20' 

.3346 

.9579 

.5 

.6 

10' 

10.3313 

9.9573 

.6 

65° 0' 

.3280 

.9567 

50' 

.3248 

.9561 

.6 

40' 

.3215 

.9555 

.6 

30' 

.3183 

.9549 

.6 

20' 

.3150 

.9543 

.6 

10' 

10.3118 

9.9537 

.u 

.7 

64° 0' 

.3086 

.9530 

50' 

.3054 

.9524 

.6 

40' 

.3023 

.9518 

.6 

30' 

.2991 

.9512 

.6 

20' 

.2960 

.9505 

.7 

10' 

10.2928 

9.9499 

• U 

.7 

63° 0' 

.2897 

.9492 

50' 

.2866 

.9486 

.6 

40' 

.2835 

.9479 

.7 

30' 

.2804 

.9473 

.6 

20' 

.2774 

.9466 

.7 

# 7 

10' 

10.2743 

9.9459 

.6 

62° 0' 

.2713 

.9453 


50' 

.2683 

.9446 

.7 

40' 

.2652 

.9439 

.7 

30' 

.2622 

.9432 

.7 

20' 

.2592 

.9425 

.7 

.7 

10' 

10.2562 

9.9418 

61° 0' 

L Tan 

L Sin 

dl' 

Angle 









































































16 


FOUR-PLACE LOGARITHMS OF FUNCTIONS 


IV 


Angle 


29° O' 

10 ' 

20 ' 

30' 

40' 

50' 

30° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

31° 0' 

10 ' 

20 ' 

* 30' 

40' 
50' 

32° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

33° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

34° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

35° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

36° 0' 

10 ' 

20 ' 

30' 

40' 

50' 

37° 0' 


L Sin 


9.6856 


d 1' 


.6878 

.6901 

.6923 

.6946 

.6968 

9.6990 

.7012 

.7033 

.7055 

.7076 

.7097 

9.7118 

.7139 

.7160 

.7181 

.7201 

.7222 

9.7242 

.7262 

.7282 

.7302 

.7322 

.7342 

9.7361 

.7380 

.7400 

.7419 

.7438 

.7457 

9.7476 

.7494 

.7513 

.7531 

.7550 

.7568 

9.7586 

.7604 

.7622 

.7640 

.7657 

.7675 

9.7692 

.7710 

.7727 

.7744 

.7761 

.7778 


9.7795 


2.2 

2.3 

2.2 

2.3 

2.2 

2.2 

2.2 

2.1 

2.2 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.0 

2.1 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

1.9 

1.9 

2.0 

1.9 

1.9 

1.9 

1.9 

1.8 

1.9 

1.8 

1.9 

1.8 

1.8 

1.8 

1.8 

1.8 

1.7 

1.8 

1.7 

1.8 

1.7 

1.7 

1.7 

1.7 

1.7 


L Cos 


dl' 


L Tan 

c d 1' 

L Cot 

L Cos 

9.7438 

2.9 

10.2562 

9.9418 

.7467 

.2533 

.9411 

.7497 

3.0 

.2503 

.9404 

.7526 

2.9 

.2474 

.9397 

.7556 

3.0 

.2444 

.9390 

.7585 

2.9 

O ft 

.2415 

.9383 

9.7614 

o n 

10.2386 

9.9375 

.7644 

o.U 

2.9 

.2356 

.9368 

.7673 

.2327 

.9361 

.7701 

2.8 

.2299 

.9353 

.7730 

2.9 

.2270 

.9346 

.7759 

2.9 

O ft 

.2241 

.9338 

9.7788 

2.8 

10.2212 

9.9331 

.7816 

.2184 

.9323 

.7845 

2.9 

.2155 

.9315 

.7873 

2.8 

.2127 

.9308 

.7902 

2.9 

.2098 

.9300 

.7930 

2.8 

2.8 

2.8 

.2070 

.9292 

9.7958 

10.2042 

9.9284 

.7986 

.2014 

.9276 

.8014 

2.8 

.1986 

.9268 

.8042 

2.8 

.1958 

.9260 

.8070 

2.8 

.1930 

.9252 

.8097 

2.7 

2.8 

2.8 

.1903 

.9244 

9.8125 

10.1875 

9.9236 

.8153 

.1847 

.9228 

.8180 

2.7 . 

.1820 

.9219 

.8208 

2.8 

.1792 

.9211 

.8235 

2.7 

.1765 

.9203 

.8263 

2.8 

2.7 

2.7 

.1737 

.9194 

9.8290 

10.1710 

9.9186 

.8317 

.1683 

.9177 

.8344 

2.7 

.1656 

.9169 

.8371 

2.7 

.1629 

.9160 

.8398 

2.7 

.1602 

.9151 

.8425 

2.7 

2.7 

2.7 

.1575 

.9142 

9.8452 

10.1548 

9.9134 

.8479 

.1521 

.9125 

.8506 

2.7 

.1494 

.9116 

.8533 

2.7 

.1467 

.9107 

.8559 

2.6 

.1441 

.9098 

.8586 

2.7 

2.7 

2.6 

.1414 

.9089 

9.8613 

10.1387 

9.9080 

.8639 

.1361 

.9070 

.8666 

2.7 

.1334 

.9061 

.8692 

2.6 

.1308 

.9052 

.8718 

2.6 

.1282 

.9042 

.8745 

2.7 

2.6 

.1255 

.9033 

9.8771 

10.1229 

9.9023 

L Cot 

c d 1' 

L Tan 

L Sin 


dl' 


.8 

.8 

.8 

.8 

.8 

.8 

.9 

.8 

.8 

.9 

.8 

.9 

.8 

.9 

.9 

.9 

.8 

.9 

.9 

.9 

.9 

.9 

.9 

1.0 

.9 

.9 

1.0 

.9 

1.0 


dl' 


61° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

60° 0' 

50' 

40' 

30' 

20 ' 

10 ', 

59° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

58° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

57° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

56° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

55° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

54° 0' 

50' 

40' 

30' 

20 ' 

10 ' 

53° 0' 


Angle 






















































































































IV 


FOUR-PLACE LOGARITHMS OF FUNCTIONS 


17 


Angle 

L Sin 

d 1' 

L Tan 

c d 1' 

L Cot 

L Cos 

d 1' 


37° O' 

9.7795 

1.6 

9.8771 

2.6 

10.1229 

9.9023 

.9 

53° O' 

10' 

.7811 

.8797 

.1203 

.9014 

50' 

20' 

.7828 

1.7 

.8824 

2.7 

.1176 

.9004 

1.0 

40' 

30' 

.7844 

1.6 

.8850 

2.6 

.1150 

.8995 

.9 

30' 

40' 

.7861 

1.7 

.8876 

2.6 

.1124 

.8985 

1.0 

20' 

50' 

.7877 

1.6 

1.6 

1.7 

1.6 

.8902 

2.6 

2.6 

2.6 

2.6 

.1098 

.8975 

1.0 

i.o 

1.0 

10' 

to 

00 

o 

o 

9.7893 

9.8928 

10.1072 

9.8965 

52° 0' 

10' 

.7910 

.8954 

.1046 

.8955 

50' 

20' 

.7926 

.8980 

.1020 

.8945 

1.0 

40' 

30' 

.7941 

1.5 

.9006 

2.6 

.0994 

.8935 

1.0 

30' 

40' 

.7957 

1.6 

.9032 

2.6 

.0968 

.8925 

1.0 

20' 

50' 

.7973 

1.6 

1.6 

1.5 

1.6 

.9058 

2.6 

2.6 

2.6 

2.5 

.0942 

.8915 

1.0 

1.0 

1.0 

10' 

39° 0' 

9.7989 

9.9084 

10.0916 

9.8905 

51° 0' 

10' 

.8004 

.9110 

.0890 

.8895 

50' 

20' 

.8020 

.9135 

.0865 

.8884 

i.i 

40' 

30' 

.8035 

1.5 

.9161 

2.6 

.0839 

.8874 

1.0 

30' 

40' 

.8050 

1.5 

.9187 

2.6 

.0813 

.8864 

1.0 

20' 

50' 

.8066 

1.6 

1.5 

1.5 

1.5 

.9212 

2.5 

2.6 

2.6 

2.5 

.0788 

.8853 

1.1 

1.0 

1.1 

10' 

o 

o 

o 

9.8081 

9.9238 

10.0762 

9.8843 

50° O' 

10' 

.8096 

.9264 

.0736 

.8832 

50' 

20' 

.8111 

.9289 

.0711 

.8821 

1.1 

40' 

30' 

.8125 

1.4 

.9315 

2.6 

.0685 

.8810 

1.1 

30' 

40' 

.8140 

1.5 

.9341 

2.6 

.0659 

.8800 

1.0 

20' 

50' 

.8155 

1.5 

1.4 

1.5 

1.4 

.9366 

2.5 

2.6 

2.5 

.0634 

.8789 

1.1 

1.1 

1.1 

10' 

£*■ 

H*' 

O 

© 

9.8169 

9.9392 

10.0608 

9.8778 

49° 0' 

10' 

.8184 

.9417 

.0583 

.8767 

50' 

20' 

.8198 

.9443 

2.6 

.0557 

.8756 

1.1 

40' 

30' 

.8213 

1.5 

.9468 

2.5 

.0532 

.8745 

1.1 

30' 

40' 

.8227 

1.4 

.9494 

2.6 

.0506 

.8733 

1.2 

20' 

50' 

.8241 

1.4 

1.4 

1.4 

1.4 

.9519 

2.5 

2.5 

2.6 

.0481 

.8722 

1.1 

1.1 

1.2 

1.1 

10' 

s 

o 

© 

9.8255 

9.9544 

10.0456 

9.8711 

o 

o 

00 

rn 

10' 

.8269 

.9570 

.0430 

.8699 

50' 

20' 

.8283 

.9595 

2.5 

.0405 

.8688 

40' 

30' 

.8297 

1.4 

.9621 

2.6 

.0379 

.8676 

1.2 

30' 

40' 

.8311 

1.4 

.9646 

2.5 

.0354 

.8665 

1.1 

20' 

50' 

.8324 

1.3 

1.4 

1.3 

.9671 

2.5 

2.6 

2.5 

.0329 

.8653 

1.2 

1.2 

1.2 

1.1 

10' 

43° 0' 

9.8338 

9.9697 

10.0303 

9.8641 

47° 0' 

10' 

.8351 

.9722 

.0278 

.8629 

50' 

20' 

.8365 

1.4 

.9747 

2.5 

.0253 

.8618 

40' 

30' 

.8378 

1.3 

.9772 

2.5 

.0228 

.8606 

1.2 

30' 

40' 

.8391 

1.3 

.9798 

2.6 

.0202 

.8594 

1.2 

20' 

50' 

.8405 

1.4 

1.3 

1.3 

.9823 

2.5 

2.5 

2.6 

2.5 

! .0177 

.8582 

1.2 

1.3 

1.2 

1.2 

10' 

o 

© 

9.8418 

9.9848 

10.0152 

9.8569 

46° 0' 

10' 

.8431 

.9874 

.0126 

.8557 

50' 

20' 

.8444 

1.3 

.9899 

.0101 

.8545 

40' 

30' 

.8457 

1.3 

.9924 

2.5 

.0076 

.8532 

1.3 

30' 

40' 

.8469 

1.2 

.9949 

2.5 

.0051 

.8520 

1.2 

20' 

50' 

.8482 

1.3 

1.3 

.9975 

2.6 

2.5 

.0025 

.8507 

1.3 

1.2 

10' 

45° 0' 

9.8495 

10.0000 

10.0000 

9.8495 

45° 0' 


L Cos 

d 1' 

L Cot 

c d 1' 

L Tan 

L Sin 

d 1' 

Angle 




























































































FIVE-PLACE TABLES 








TABLE V 


FIVE-PLACE LOGARITHMS 
OF THE 

TRIGONOMETRIC FUNCTIONS 
OF 

ANGLES BETWEEN 0° AND 3° 
AND BETWEEN 87° AND 90° 


Note. — For angles between 0° and 3° and between 87° and 90° Table Va or Table Vb may 
be used to avoid interpolation in Table IV or in ordinary five-place tables; the results thus 
obtained are more accurate. Errors of interpolation in Table Vb correspond to differences 
of angle of less than 1 "; Table Vo gives still more accurate results. 


Va. AUXILIARY TABLE OF S AND T FOR A IN MINUTES 

For angles near 0°: log sin A = S + log A' and log tan A = T -f- log A'. 

For angles near 90°: log cos A = Si + log A'i and log cot A = Ti + log A'i where A\ 

is the number of minutes in 90° — A and Si and Ti are corresponding values of S and T. 


A' 

S + 10 


A' 

T + 10 

A' 

T + 10 

0' - 13' 

6.46 373 


0' - 26' 

6.46 373 

131' - 133' 

6.46 394 

14' - 42' 

372 


27' -39' 

374 

134' - 136' 

395 

43' - 58' 

371 


40' - 48' 

375 

137' - 139' 

396 

59' - 71' 

6.46 370 


49' - 56' 

6.46 376 

140' - 142' 

6.46 397 

72' - 81' 

369 


57' - 63' 

377 

143' - 145' 

398 

82' - 91' 

368 


64' - 69' 

378 

146' - 148' 

399 

92' - 99' 

6.46 367 


70' - 74' 

6.46 379 

149' - 150' 

6.46 400 

100' - 107' 

366 


75' - 80' 

380 

151' - 153' 

401 

108' - 115' 

365 


81' - 85' 

381 

154' - 156' 

402 

116' - 121' 

6.46 364 


86' - 89' 

6.46 382 

157' - 158' 

6.46 403 

122' - 128' 

363 


90' - 94' 

383 

159' - 161' 

404 

129' - 134' 

362 


95' - 98' 

384 

162' - 163' 

405 

135' - 140' 

6.46 361 


99' - 102' 

6.46 385 

164' - 166' 

6.46 406 

141' - 146' 

360 


103' - 106' 

386 

167' - 168' 

407 

147' - 151' 

359 


107' - 110' 

387 

169' - 171' 

408 

152' - 157' 

6.46 358 


111' - 113' 

6.46 388 

172' - 173' 

6.46 409 

158' - 162' 

357 


114' - 117' 

389 

174' - 175' 

410 

163' - 167' 

356 


118' - 120' 

390 

176' - 178' 

411 

168' - 171' 

6.46 355 


121' - 124' 

6.46 391 

179' - 180' 

6.46 412 

172' - 176' 

354 


125' - 127' 

392 

181' - 182' 

413 

177' - 181' 

353 

| 128' - 130' 

393 

1 

183' - 184' 

414 


21 



















22 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


Vb 


ANGLES NEAR 0° AND 90° 


Angle 

Log Sin A + 10 

C 


A 

0" 

10" 

20" 

30" 

40" 

50" 

.0000 


0° 0' 


5.68 557 

5.98 660 

6.16 270 

6.28 763 

6.38 454 

0 


1' 

6.46 373 

6.53 067 

6.58 866 

6.63 982 

6.68 557 

6.72 697 

0 


2' 

6.76 476 

6.79 952 

6.83 170 

6.86 167 

6.88 969 

6.91 602 

0 


3' 

6.94 085 

6.96 433 

6.98 660 

7.00 779 

7.02 800 

7.04 730 

0 

H 1 

4' 

7.06 579 

7.08 351 

7.10 055 

7.11 694 

7.13 273 

7.14 797 

0 


5' 

7.16 270 

7.17 694 

7.19 072 

7.20 409 

7.21 705 

7.22 964 

0 

“o 

6' 

7.24 188 

7.25 378 

7.26 536 

7.27 664 

7.28 763 

7.29 836 

0 

•* 73 

O O 

7' 

7.30 882 

7.31 904 

7.32 903 

7.33 879 

7.34 833 

7.35 767 

0 

t-H © 

8' 

7.36 682 

7.37 577 

7.38 454 

7.39 314 

7.40 158 

7.40 985 

0 

1 II 

9' 

7.41 797 

7.42 594 

7.43 376 

7.44 145 

7.44 900 

7.45 643 

0 

OO) 

0° 10' 

7.46 373 

7.47 090 

7.47 797 

7.48 491 

7.49 175 

7.49 849 

0 

1.9 

11' 

7.50 512 

7.51 165 

7.51 808 

7.52 442 

7.53 067 

7.53 683 

0 


12' 

7.54 291 

7.54 890 

7.55 481 

7.56 064 

7.56 639 

7.57 206 

0 

II rn 

13' 

7.57 767 

7.58 320 

7.58 866 

7.59 406 

7.59 939 

7.60 465 

0 

ii g 

14' 

7.60 985 

7.61 499 

7.62 007 

7.62 509 

7.63 006 

7.63 496 

1 

^.2 

15' 

7.63 982 

7.64 461 

7.64 936 

7.65 406 

7.65 870 

7.66 330 

1 

8 1 

16' 

7.66 784 

7.67 235 

7.67 680 

7.68 121 

7.68 557 

7.68 989 

1 

° S 

17' 

7.69 417 

7.69 841 

7.70 261 

7.70 676 

7.71 088 

7.71 496 

1 

$2 

18' 

7.71 900 

7.72 300 

7.72 697 

7.73 090 

7.73 479 

7.73 865 

1 

M +» 

19' 

7.74 248 

7.74 627 

7.75 003 

7.75 376 

7.75 745 

7.76 112 

1 

© 

• “ 73 

o id 

0° 20' 

7.76 475 

7.76 836 

7.77 193 

7.77 548 

7.77 899 

7.78 248 

1 

7o 

21' 

7.78 594 

7.78 938 

7.79 278 

7.79 616 

7.79 952 

7.80 284 

1 

1 03 

22' 

7.80 615 

7.80 942 

7.81 268 

7.81 591 

7.81 911 

7.82 229 

1 


23' 

7.82 545 

7.82 859 

7.83 170 

7.83 479 

7.83 786 

7.84 091 

1 

g d 

24' 

7.84 393 

7.84 694 

7.84 992 

7.85 289 

7.85 583 

7.85 876 

1 

c3 

■PO 

25' 

7.86 166 

7.86 455 

7.86 741 

7.87 026 

7.87 309 

7.87 590 

1 

IfSoas 

26' 

7.87 870 

7.88 147 

7.88 423 

7.88 697 

7.88 969 

7.89 240 

1 

A a , 

27' 

7.89 509 

7.89 776 

7.90 041 

7.90 305 

7.90 568 

7.90 829 

1 

i § 

28' 

7.91 088 

7.91 346 

7.91 602 

7.91 857 

7.92 110 

7.92 362 

1 

O -Eo 

29' 

7.92 612 

7.92 861 

7.93 108 

7.93 354 

7.93 599 

7.93 842 

2 

H OO 

0° 30' 

7.94 084 

7.94 325 

7.94 564 

7.94 802 

7.95 039 

7.95 274 

2 

II 73 
rrl ®‘® 

31' 

7.95 508 

7.95 741 

7.95 973 

7.96 203 

7.96 432 

7.96 660 

2 

7: If II 

32' 

7.96 887 

7.97 113 

7.97 337 

7.97 560 

7.97 782 

7.98 003 

2 


33' 

7.98 223 

7.98 442 

7.98 660 

7.98 876 

7.99 092 

7.99 306 

2 


34' 

7.99 520 

7.99 732 

7.99 943 

8.00 154 

8.00 363 

8.00 571 

2 


35' 

8.00 779 

8.00 985 

8.01 190 

8.01 395 

8.01 598 

8.01 801 

2 

HgU 

36' 

8.02 002 

8.02 203 

8.02 402 

8.02 601 

8.02 799 

8.02 996 

2 


37' 

8.03 192 

8.03 387 

8.03 581 

8.03 775 

8.03 967 

8.04 159 

3 


38' 

8.04 350 

8.04 540 

8.04 729 

8.04 918 

8.05 105 

8.05 292 

3 

is i 

39' 

8.05 478 

8.05 663 

8.05 848 

8.06 031 

8.06 214 

8.06 396 

3 


0° 40' 

8.06 578 

8.06 758 

8.06 938 

8.07 117 

8.07 295 

8.07 473 

3 

03 w 

41' 

8.07 650 

8.07 826 

8.08 002 

8.08 176 

8.08 350 

8.08 524 

3 

m . d 

42' 

8.08 696 

8.08 868 

8.09 040 

8.09 210 

8.09 380 

8.09 550 

4 

9 © c3 
|_1 o -u 

43' 

8.09 718 

8.09 886 

8.10 054 

8.10 220 

8.10 386 

8.10 552 

4 

n — 

44' 

8.10 717 

8.10 881 

8.11 044 

8.11 207 

8.11 370 

8.11 531 

4 

ii "2, ii 

45' 

8.11 693 

8.11 853 

8.12 013 

8.12 172 

8.12 331 

8.12 489 

4 


46' 

8.12 647 

8.12 804 

8.12 961 

8.13 117 

8.13 272 

8.13 427 

4 

g 

47' 

8.13 581 

8.13 735 

8.13 888 

8.14 041 

8.14 193 

8.14 344 

4 


48' 

8.14 495 

8.14 646 

8.14 796 

8.14 945 

8.15 094 

8.15 243 

4 


49' 

8.15 391 

8.15 538 

8.15 685 

8.15 832 

8.15 978 

8.16 123 

4 

m.Sgq 

0° 50' 

8.16 268 

8.16 413 

8.16 557 

8.16 700 

8.16 843 

8.16 986 

5 

•7 1 

51' 

8.17 128 

8.17 270 

8.17 411 

8.17 552 

8.17 692 

8.17 832 

5 

£ 

52' 

8.17 971 

8.18 110 

8.18 249 

8.18 387 

8.18 524 

8.18 662 

5 

ftgs 

53' 

8.18 798 

8.18 935 

8.19 071 

8.19 206 

8.19 341 

8.19 476 

5 

© © o 

54' 

8.19 610 

8.19 744 

8.19 877 

8.20 010 

8.20 143 

8.20 275 

6 

-g © o 

55' 

8.20 407 

8.20 538 

8.20 669 

8.20 800 

8.20 930 

8.21 060 

6 

£ a ii 

56' 

8.21 189 

8.21 319 

8.21 447 

8.21 576 

8.21 703 

8.21 831 

6 


57' 

8.21 958 

8.22 085 

8.22 211 

8.22 337 

8.22 463 

8.22 588 

6 


58' 

8.22 713 

8.22 838 

8.22 962 

8.23 086 

8.23 210 

8.23 333 

6 

cj * 

59' 

8.23 456 

8.23 578 

8.23 700 

8.23 822 

8.23 944 

8.24 065 

6 






















































Vb 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


23 


ANGLES NEAR 0° AND 90° 


Angle 

Log Sin A + 10 

C 


A 

0" 

10" 

20" 

30" 

40" 

50" 

.0000 


1° O' 

8.24 186 

8.24 306 

8.24 426 

8.24 546 

8.24 665 

8.24 785 

7 


1' 

8.24 903 

8.25 022 

8.25 140 

8.25 258 

8.25 375 

8.25 493 

7 


2' 

8.25 609 

8.25 726 

8.25 842 

8.25 958 

8.26 074 

8.26 189 

7 


3' 

8.26 304 

8.26 419 

8.26 533 

8.26 648 

8.26 761 

8.26 875 

7 

&l 

4' 

8.26 988 

8.27 101 

8.27 214 

8.27 326 

8.27 438 

8.27 550 

8 

go 

5' 

8.27 661 

8.27 773 

8.27 883 

8.27 994 

8.28 104 

8.28 215 

8 


6' 

8.28 324 

8.28 434 

8.28 543 

8.28 652 

8.28 761 

8.28 869 

8 

GO 

O q 

r 

8.28 977 

8.29 085 

8.29 193 

8.29 300 

8.29 407 

8.29 514 

9 

r-i 0 

8' 

8.29 621 

8.29 727 

8.29 833 

8.29 939 

8.30 044 

8.30 150 

9 

1 II 

9' 

8.30 255 

8.30 359 

8.30 464 

8.30 568 

8.30 672 

8.30 776 

9 


1° 10' 

8.30 879 

8.30 983 

8.31 086 

8.31 188 

8.31 291 

8.31 393 

9 

1 

11' 

8.31 495 

8.31 597 

8.31 699 

8.31 800 

8.31 901 

8.32 002 

9 

o’® 

12' 

8.32 103 

8.32 203 

8.32 303 

8.32 403 

8.32 503 

8.32 602 

.00010 


13' 

8.32 702 

8.32 801 

8.32 899 

8.32 998 

8.33 096 

8.33 195 

10 

11 fl 

14' 

8.33 292 

8.33 390 

8.33 488 

8.33 585 

8.33 682 

8.33 779 

10 


15' 

8.33 875 

8.33 972 

8.34 068 

8.34 164 

8.34 260 

8.34 355 

10 

8-a 

16' 

8.34 450 

8.34 546 

8.34 640 

8.34 735 

8.34 830 

8.34 924 

11 

o 2 

17' 

8.35 018 

8.35 112 

8.35 206 

8.35 299 

8.35 392 

8.35 485 

11 

M <v 

0^3 

18' 

8.35 578 

8.35 671 

8.35 764 

8.35 856 

8.35 948 

8.36 040 

11 


19' 

8.36 131 

8.36 223 

8.36 314 

8.36 405 

8.36 496 

8.36 587 

12 

0 

GO 

1° 20' 

8.36 678 

8.36 768 

8.36 858 

8.36 948 

8.37 038 

8.37 128 

12 

T "“ l O 
• f—> 

21' 

8.37 217 

8.37 306 

8.37 395 

8.37 484 

8.37 573 

8.37 662 

12 

1 05 

22' 

8.37 750 

8.37 838 

8.37 926 

8.38 014 

8.38 101 

8.38 189 

12 

^'9 

23' 

8.38 276 

8.38 363 

8.38 450 

8.38 537 

8.38 624 

8.38 710 

13 

Cl 
a os 

24' 

8.38 796 

8.38 882 

8.38 968 

8.39 054 

8.39 139 

8.39 225 

13 

-2° 

25' 

8.39 310 

8.39 395 

8.39 480 

8.39 565 

8.39 649 

8.39 734 

13 


26' 

8.39 818 

8.39 902 

8.39 986 

8.40 070 

8.40 153 

8.40 237 

14 

>3 §7 

27' 

8.40 320 

8.40 403 

8.40 486 

8.40 569 

8.40 651 

8.40 734 

14 

1 g 

28' 

8.40 816 

8.40 898 

8.40 980 

8.41 062 

8.41 144 

8.41 225 

15 

'lb 

29' 

8.41 307 

8.41 388 

8.41 469 

8.41 550 

8.41 631 

8.41 711 

15 

O 5)05 

1° 30' 

8.41 792 

8.41 872 

8.41 952 

8.42 032 

8.42 112 

8.42 192 

15 

II ® >=l 
,2’3 

31' 

8.42 272 

8.42 351 

8.42 430 

8.42 510 

8.42 589 

8.42 667 

15 

^ C II 

32' 

8.42 746 

8.42 825 

8.42 903 

8.42 982 

8.43 060 

8.43 138 

16 

o ^ 

33' 

8.43 216 

8.43 293 

8.43 371 

8.43 448 

8.43 526 

8.43 603 

16 


34' 

8.43 680 

8.43 757 

8.43 834 

8.43 910 

8.43 987 

8.44 063 

16 

osg 

35' 

8.44 139 

8.44 216 

8.44 292 

8.44 367 

8.44 443 

8.44 519 

17 


36' 

8.44 594 

8.44 669 

8.44 745 

8.44 820 

8.44 895 

8.44 969 

17 

•--3 •- 

37' 

8.45 044 

8.45 119 

8.45 193 

8.45 267 

8.45 341 

8.45 415 

17 


38' 

8.45 489 

8.45 563 

8.45 637 

8.45 710 

8.45 784 

8.45 857 

18 

+5 l 

39' 

8.45 930 

8.46 003 

8.46 076 

8.46 149 

8.46 222 

8.46 294 

18 

^ o° 

1° 40' 

8.46 366 

8.46 439 

8.46 511 

8.46 583 

8.46 655 

8.46 727 

18 


41' 

8.46 799 

8.46 870 

8.46 942 

8.47 013 

8.47 084 

8.47 155 

19 

M . 5 

42' 

8.47 226 

8.47 297 

8.47 368 

8.47 439 

8.47 509 

8.47 580 

19 

o 03 

43' 

8.47 650 

8.47 720 

8.47 790 

8.47 860 

8.47 930 

8.48 000 

20 

^-2 ii 

44' 

8.48 069 

8.48 139 

8.48 208 

8.48 278 

8.48 347 

8.48 416 

20 

II ’ft 

45' 

8.48 485 

8.48 554 

8.48 622 

8.48 691 

8.48 760 

8.48 828 

20 


46' 

8.48 896 

8.48 965 

8.49 033 

8.49 101 

8.49 169 

8.49 236 

20 

rt—< o 

47' 

8.49 304 

8.49 372 

8.49 439 

8.49 506 

8.49 574 

8.49 641 

21 

(U w 

48' 

8.49 708 

8.49 775 

8.49 842 

8.49 908 

8.49 975 

8.50 042 

21 


49' 

8.50 108 

8.50 174 

8.50 241 

8.50 307 

8.50 373 

8.50 439 

22 


1° 50' 

8.50 504 

8.50 570 

8.50 636 

8.50 701 

8.50 767 

8.50 832 

23 

1 

51' 

8.50 897 

8.50 963 

8.51 028 

8.51 092 

8.51 157 

8.51 222 

23 

cq °o 

52' 

8.51 287 

8.51 351 

8.51 416 

8.51 480 

8.51 544 

8.51 609 

23 

ftg- 

53' 

8.51 673 

8.51 737 

8.51 801 

8.51 864 

8.51 928 

8.51 992 

23 

u®o 

54' 

8.52 055 

8.52 119 

8.52 182 

8.52 245 

8.52 308 

8.52 371 

24 

0,2 “ 

55' 

8.52 434 

8.52 497 

8.52 560 

8.52 623 

8.52 685 

8.52 748 

24 

££ II 

56' 

8.52 810 

8.52 872 

8.52 935 

8.52 997 

8.53 059 

8.53 121 

25 

* 

57' 

8.53 183 

8.53 245 

8.53 306 

8.53 368 

8.53 429 

8.53 491 

25 

™ S,fl 

58' 

8.53 552 

8.53 614 

8.53 675 

8.53 736 

8.53 797 

8.53 858 

26 

- c3 

03 -+-S 

59' 

8.53 919 

8.53 979 

8.54 040 

8.54 101 

8.54 161 

8.54 222 

26 




























































24 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


Vb 


ANGLES NEAR 0° AND 90° 


Angle 

Log Sin A + 10 

C 

A 

0" 

10" 

20" 

30" 

40" 

50" 

.000 

2° O' 

8.54 282 

8.54 342 

8.54 402 

8.54 462 

8.54 522 

8.54 582 

27 

J/ 

8.54 642 

8.54 702 

8.54 762 

8.54 821 

8.54 881 

8.54 940 

27 

2' 

8.54 999 

8.55 059 

8.55 118 

8.55 177 

8.55 236 

8.55 295 

28 

3' 

8.55 354 

8.55 413 

8.55 471 

8.55 530 

8.55 589 

8.55 647 

28 

4' 

8.55 705 

8.55 764 

8.55 822 

8.55 880 

8.55 938 

8.55 996 

29 

5' 

8.56 054 

8.56 112 

8.56 170 

8.56 227 

8.56 285 

8.56 342 

29 

6' 

8.56 400 

8.56 457 

8.56 515 

8.56 572 

8.56 629 

8.56 686 

29 

7' 

8.56 743 

8.56 800 

8.56 857 

8.56 914 

8.56 970 

8.57 027 

30 

8' 

8.57 084 

8.57 140 

8.57 196 

8.57 253 

8.57 309 

8.57 365 

30 

9' 

8.57 421 

8.57 477 

8.57 533 

8.57 589 

8.57 645 

8.57 701 

31 

2° 10' 

8.57 757 

8.57 812 

8.57 868 

8.57 927 

8.57 979 

8.58 034 

31 

11' 

8.58 089 

8.58 144 

8.58 200 

8.58 255 

8.58 310 

8.58 364 

32 

12' 

8.58 419 

8.58 474 

8.58 529 

8.58 583 

8.58 638 

8.58 693 

32 

13' 

8.58 747 

8.58 801 

8.58 856 

8.58 910 

8.58 964 

8.59 018 

33 

14' 

8.59 072 

8.59 126 

8.59 180 

8.59 234 

8.59 288 

8.59 341 

33 

15' 

8.59 395 

8.59 448 

8.59 502 

8.59 555 

8.59 609 

8.59 662 

34 

16' 

8.59 715 

8.59 768 

8.59 821 

8.59 874 

8.59 927 

8.59 980 

35 

17' 

8.60 033 

8.60 086 

8.60 139 

8.60 191 

8.60 244 

8.60 296 

35 

18' 

8.60 349 

8.60 401 

8.60 454 

8.60 506 

8.60 558 

8.60 610 

35 

19' 

8.60 662 

8.60 714 

8.60 766 

8.60 818 

8.60 870 

8.60 922 

36 

2° 20' 

8.60 973 

8.61 025 

8.61 077 

8.61 128 

8.61 180 

8.61 231 

36 

21' 

8.61 282 

8.61 334 

8.61 385 

8.61 436 

8.61 487 

8.61 538 

37 

22' 

8.61 589 

8.61 640 

8.61 691 

8.61 742 

8.61 792 

8.61 843 

37 

23' 

8.61 894 

8.61 944 

8.61 995 

8.62 045 

8.62 096 

8.62 146 

38 

24' 

8.62 196 

8.62 246 

8.62 297 

8.62 347 

8.62 397 

8.62 447 

38 

25' 

8.62 497 

8.62 546 

8.62 596 

8.62 646 

8.62 696 

8.62 745 

39 

26' 

8.62 795 

8.62 844 

8.62 894 

8.62 943 

8.62 993 

8.63 042 

39 

27' 

8.63 091 

8.63 140 

8.63 189 

8.63 238 

8.63 288 

8.63 336 

40 

28' 

8.63 385 

8.63 434 

8.63 483 

8.63 532 

8.63 580 

8.63 629 

41 

29' 

8.63 678 

8.63 726 

8.63 775 

8.63 823 

8.63 871 

8.63 920 

41 

2° 30' 

8.63 968 

8.64 016 

8.64 064 

8.64 112 

8.64 160 

8.64 208 

42 

31' 

8.64 256 

8.64 304 

8.64 352 

8.64 400 

8.64 448 

8.64 495 

42 

32' 

8.64 543 

8.64 590 

8.64 638 

8.64 685 

8.64 733 

8.64 780 

43 

33' 

8.64 827 

8.64 875 

8.64 922 

8.64 969 

8.65 016 

8.65 063 

43 

34' 

8.65 110 

8.65 157 

8.65 204 

8.65 251 

8.65 298 

8.65 344 

44 

35' 

8.65 391 

8.65 438 

8.65 484 

8.65 531 

8.65 577 

8.65 624 

44 

36' 

8.65 670 

8.65 717 

8.65 763 

8.65 809 

8.65 855 

8.65 901 

45 

37' 

8.65 947 

8.65 994 

8.66 040 

8.66 085 

8.66 131 

8.66 177 

46 

38' 

8.66 223 

8.66 269 

8.66 314 

8.66 360 

8.66 406 

8.66 451 

46 

39' 

8.66 497 

8.66 542 

8.66 588 

8.66 633 

8.66 678 

8.66 724 

47 

to 

0 

© 

8.66 769 

8.66 814 

8.66 859 

8.66 904 

8.66 949 

8.66 994 

47 

41' 

8.67 039 

8.67 084 

8.67 129 

8.67 174 

8.67 219 

8.67 263 

48 

42' 

8.67 308 

8.67 353 

8.67 397 

8.67 442 

8.67 486 

8.67 531 

48 

43' 

8.67 575 

8.67 619 

8.67 664 

8.67 708 

8.67 752 

8.67 796 

49 

44' 

8.67 841 

8.67 885 

8.67 929 

8.67 973 

8.68 017 

8.68 060 

49 

45' 

8.68 104 

8.68 148 

8.68 192 

8.68 236 

8.68 279 

8.68 323 

50 

46' 

8.68 367 

8.68 410 

8.68 454 

8.68 497 

8.68 540 

8.68 584 

51 

47' 

8.68 627 

8.68 670 

8.68 714 

8.68 757 

8.68 800 

8.68 843 

51 

48' 

8.68 886 

8.68 929 

8.68 972 

8.69 015 

8.69 058 

8.69 101 

52 

49' 

8.69 144 

8.69 187 

8.69 229 

8.69 272 

8.69 315 

8.69 357 

53 

2° 50' 

8.69 400 

8.69 442 

8.69 485 

8.69 527 

8.69 570 

8.69 612 

53 

51' 

8.69 654 

8.69 697 

8.69 739 

8.69 781 

8.69 823 

8.69 865 

54 

52' 

8.69 907 

8.69 949 

8.69 991 

8.70 033 

8.70 075 

8.70 117 

55 

53' 

8.70 159 

8.70 201 

8.70 242 

8.70 284 

8.70 326 

8.70 367 

55 

54' 

8.70 409 

8.70 451 

8.70 492 

8.70 534 

8.70 575 

8.70 616 

56 

55' 

8.70 658 

8.70 699 

8.70 740 

8.70 781 

8.70 823 

8.70 864 

56 

56' 

8.70 905 

8.70 946 

8.70 987 

8.71 028 

8.71 069 

8.71 110 

57 

57' 

8.71 151 

8.71 192 

8.71 232 

8.71 273 

8.71 314 

8.71 355 

58 

58' 

8.71 395 

8.71 436 

8.71 476 

8.71 517 

8.71 557 

8.71 598 

58 

59' 

8.71 638 

8.71 679 

8.71 719 

8.71 759 

8.71 800 

8.71 840 

59 


See Note, p. 21. Log tan A — Log sin A + C ; Log cot A = 10 — Log tan A — 10; Log cos A = 10 — C — 10 ; except for 
a possible error of 1 in the last place. For functions of angles between 87° and 90° use the relations: sin B = cos (90° — B) ; 
tan B = cot(90° — B) ; cot B = tan(90° — B) ; cos B = sin (90° — B). 
































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


25 


0 ° 


/ 

L Sin 

L Tan 

L Cot 

L Cos 


Prop. Pts. 

0 

— 

— 

_ 

10.00 000 

60 


1 

6.46 373 

6.46 373 

13.53 627 

10.00 000 

59 


2 

6.76 476 

6.76 476 

13.23 524 

10.00 000 

58 


3 

6.94 085 

6.94 085 

13.05 915 

10.00 000 

57 


4 

7.06 579 

7.06 579 

12.93 421 

10.00 000 

56 


5 

7.16 270 

7.16 270 

12.83 730 

10.00 000 

55 


6 

7.24 188 

7.24 188 

12.75 812 

10.00 000 

54 


7 

7.30 882 

7.30 882 

12.69 118 

10.00 000 

53 


8 

7.36 682 

7.36 682 

12.63 318 

10.00 000 

52 


9 

7.41 797 

7.41 797 

12.58 203 

10.00 000 

51 


10 

7.46 373 

7.46 373 

12.53 627 

10.00 000 

50 


11 

7.50 512 

7.50 512 

12.49 488 

10.00 000 

49 


12 

7.54 291 

7.54 291 

12.45 709 

10.00 000 

48 


13 

7.57 767 

7.57 767 

12.42 233 

10.00 000 

47 


14 

7.60 985 

7.60 986 

12.39 014 

10.00 000 

46 


15 

7.63 982 

7.63 982 

12.36 018 

10.00 000 

45 


16 

7.66 784 

7.66 785 

12.33 215 

10.00 000 

44 

<N 

17 

7.69 417 

7.69 418 

12.30 582 

9.99 999 

43 

& 

18 

7.71 900 

7.71 900 

12.28 100 

9.99 999 

42 

S’® 

19 

7.74 248 

7.74 248 

12.25 752 

9.99 999 

41 

/■n 

20 

7.76 475 

7.76 476 

12.23 524 

9.99 999 

40 

■S-g 

21 

7.78 594 

7.78 595 

12.21 405 

9.99 999 

39 

O M 
£ £ 

22 

7.80 615 

7.80 615 

12.19 385 

9.99 999 

38 

^ bO 

O O 

23 

7.82 545 

7.82 546 

12.17 454 

9.99 999 

37 


24 

7.84 393 

7.84 394 

12.15 606 

9.99 999 

36 

o 

25 

7.86 166 

7.86 167 

12.13 833 

9.99 999 

35 

<5 a) 

26 

7.87 870 

7.87 871 

12.12 129 

9.99 999 

34 

!>J2 

27 

7.89 509 

7.89 510 

12.10 490 

9.99 999 

33 


28 

7.91 088 

7.91 089 

12.08 911 

9.99 999 

32 


29 

7.92 612 

7.92 613 

12.07 387 

9.99 998 

31 

>S 

30 

7.94 084 

7.94 086 

12.05 914 

9.99 998 

30 

"S3 

31 

7.95 508 

7.95 510 

12.04 490 

9.99 998 

29 


32 

7.96 887 

7.96 889 

12.03 111 

9.99 998 

28 


33 

7.98 223 

7.98 225 

12.01 775 

9.99 998 

27 

0> rt 

34 

7.99 520 

7.99 522 

12.00 478 

9.99 998 

26 

B O 

35 

8.00 779 

8.00 781 

11.99 219 

9.99 998 

25 

Cl '** 

.2 ° 

36 

8.02 002 

8.02 004 

11.97 996 

9.99 998 

24 


37 

8.03 192 

8.03 194 

11.96 806 

9.99 997 

23 


38 

8.04 350 

8.04 353 

11.95 647 

9.99 997 

22 

a 2 

39 

8.05 478 

8.05 481 

11.94 519 

9.99 997 

21 


40 

8.06 578 

8.06 581 

11.93 419 

9.99 997 

20 

.S ® 

41 

8.07 650 

8.07 653 

11.92 347 

9.99 997 

19 


42 

8.08 696 

8.08 700 

11.91 300 

9.99 997 

18 

oj 

43 

8.09 718 

8.09 722 

11.90 278 

9.99 997 

17 


44 

8.10 717 

8.10 720 

11.89 280 

9.99 996 

16 

O 

45 

8.11 693 

8.11 696 

11.88 304 

9.99 996 

15 


46 

8.12 647 

8.12 651 

11.87 349 

9.99 996 

14 


47 

8.13 581 

8.13 585 

11.86 415 

9.99 996 

13 


48 

8.14 495 

8.14 500 

11.85 500 

9.99 996 

12 


49 

8.15 391 

8.15 395 

11.84 605 

9.99 996 

11 


50 

8.16 268 

8.16 273 

11.83 727 

9.99 995 

10 


51 

8.17 128 

8.17 133 

11.82 867 

9.99 995 

9 


52 

8.17 971 

8.17 976 

11.82 024 

9.99 995 

8 


53 

8.18 798 

8.18 804 

11.81 196 

9.99 995 

7 


54 

8.19 610 

8.19 616 

11.80 384 

9.99 995 

6 


55 

8.20 407 

8.20 413 

11.79 587 

9.99 994 

5 


56 

8.21 189 

8.21 195 

11.78 805 

9.99 994 

4 


57 

8.21 958 

8.21 964 

11.78 036 

9.99 994 

3 


58 

8.22 713 

8.22 720 

11.77 280 

9.99 994 

2 


59 

8.23 456 

8.23 462 

11.76 538 

9.99 994 

1 


60 

8.24 186 

8.24 192 

11.75 808 

9.99 993 

0 



L Cos 

L Cot 

L Tan 

L Sin 

' 

Prop. Pts. 


89 








































FIVE-PLACE LOGARITHMS OF FUNCTIONS 

1 ° 


VI 


26 


' 

L Sin 

L Tan 

L Cot 

L Cos. 


Prop. Pts. 

0 

8.24 186 

8.24 192 

11.75 808 

9.99 993 

60 


1 

8.24 903 

8.24 910 

11.75 090 

9.99 993 

59 


2 

8.25 609 

8.25 616 

11.74 384 

9.99 993 

58 


3 

8.26 304 

8.26 312. 

11.73 688 

9.99 993 

57 


4 

8.26 988 

8.26 996 

11.73 004 

9.99 992 

56 


5 

8.27 661 

8.27 669 

11.72 331 

9.99 992 

55 


6 

8.28 324 

8.28 332 

11.71 668 

9.99 992 

54 


7 

8.28 977 

8.28 986 

11.71 014 

9.99 992 

53 


8 

8.29 621 

8.29 629 

11.70 371 

9.99 992 

52 


9 

8.30 255 

8.30 263 

11.69 737 

9.99 991 

51 


10 

8.30 879 

8.30 888 

11.69 112 

9.99 991 

50 


11 

8.31 495 

8.31 505 

11.68 495 

9.99 991 

49 


12 

8.32 103 

8.32 112 

11.67 888 

9.99 990 

48 


13 

8.32 702 

8.32 711 

11.67 289 

9.99 990 

47 


14 

8.33 292 

8.33 302 

11.66 698 

9.99 990 

46 


15 

8.33 875 

8.33 886 

11.66 114 

9.99 990 

45 

r-4 

16 

8.34 450 

8.34 461 

11.65 539 

9.99 989 

44 


17 

8.35 018 

8.35 029 

11.64 971 

9.99 989 

43 

0) 

bJO 

18 

8.35 578 

8.35 590 

11.64 410 

9.99 989 

42 

a 

P* 

19 

8.36 131 

8.36 143 

11.63 857 

9.99 989 

41 

a> . 

20 

8.36 678 

8.36 689 

11.63 311 

9.99 988 

40 

o § 

21 

8.37 217 

8.37 229 

11.62 771 

9.99 988 

39 


22 

8.37 750 

8.37 762 

11.62 238 

9.99 988 

38 


23 

8.38 276 

8.38 289 

11.61 711 

9.99 987 

37 

m 

24 

8.38 796 

8.38 809 

11.61 191 

9.99 987 

36 

bfl 

O 

25 

8.39 310 

8.39 323 

11.60 677 

9.99 987 

35 


26 

8.39 818 

8.39 832 

11.60 168 

9.99 986 

34 

P* o 

27 

8.40 320 

8.40 334 

11.59 666 

9.99 986 

33 

O $ 

28 

8.40 816 

8.40 830 

11.59 170 

9.99 986 

32 

e-f 

29 

8.41 307 

8.41 321 

11.58 679 

9.99 985 

31 

> $ 

30 

8.41 792 

8.41 807 

11.58 193 

9.99 985 

30 

JS'd 

31 

8.42 272 

8.42 287 

11.57 713 

9.99 985 

29 


32 

8.42 746 

8.42 762 

11.57 238 

9.99 984 

28 


33 

8.43 216 

8.43 232 

11.56 768 

9.99 984 

27 


34 

8.43 680 

8.43 696 

11.56 304 

9.99 984 

26 

33 

35 

8.44 139 

8.44 156 

11.55 844 

9.99 983 

25 


36 

8.44 594 

8.44 611 

11.55 389 

9.99 983 

24 

•Sj o 

37 

8.45 044 

8.45 061 

11.54 939 

9.99 983 

23 

b3 >4-i 

38 

8.45 489 

8.45 507 

11.54 493 

9.99 982 

22 

l S 

39 

8.45 930 

8.45 948 

11.54 052 

9.99 982 

21 

<u o 

40 

8.46 366 

8.46 385 

11.53 615 

9.99 982 

20 

.2 2 

41 

8.46 799 

8.46 817 

11.53 183 

9.99 981 

19 

T3-S 

42 

8.47 226 

8.47 245 

11.52 755 

9.99 981 

18 

•3 g 

43 

8.47 650 

8.47 669 

11.52 331 

9.99 981 

17 

£-g 

44 

8.48 069 

8.48 089 

11.51 911 

9.99 980 

16 

0,3 

45 

8.48 485 

8.48 505 

11.51 495 

9.99 980 

15 


46 

8.48 896 

8.48 917 

11.51 083 

9.99 979 

14 


47 

8.49 304 

8.49 325 

11.50 675 

9.99 979 

13 


48 

8.49 708 

8.49 729 

11.50 271 

9.99 979 

12 


49 

8.50 108 

8.50 130 

11.49 870 

9.99 978 

11 


50 

8.50 504 

8.50 527 

11.49 473 

9.99 978 

10 


51 

8.50 897 

8.50 920 

11.49 080 

9.99 977 

9 


52 

8.51 287 

8.51 310 

11.48 690 

9.99 977 

8 


53 

8.51 673 

8.51 696 

11.48 304 

9.99 977 

7 


54 

8.52 055 

8.52 079 

11.47 921 

9.99 976 

6 


55 

8.52 434 

8.52 459 

11.47 541 

9.99 976 

5 


56 

8.52 810 

8.52 835 

11.47 165 

9.99 975 

4 


57 

8.53 183 

8.53 208 

11.46 792 

9.99 975 

3 


58 

8.53 552 

8.53 578 

11.46 422 

9.99 974 

2 


59 

8.53 919 

8.53 945 

11.46 055 

9.99 974 

1 


60 

8.54 282 

8.54 308 

11.45 692 

9.99 974 

0 



L Cos 

L Cot 

L Tan 

L Sin 

/ 

Prop. Pts. 


88 ; 

































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


27 


2 ° 


f 

L Sin 

L Tan 

L Cot 

L Cos 


Prop. Pts. 

0 

8.54 282 

8.54 308 

11.45 692 

9.99 

974 

60 


1 

8.54 642 

8.54 669 

11.45 331 

9.99 

973 

59 


2 

8.54 999 

8.55 027 

11.44 973 

9.99 

973 

58 


3 

8.55 354 

8.55 382 

11.44 618 

9.99 

972 

57 


4 

8.55 705 

8.55 734 

11.44 266 

9.99 

972 

56 


5 

8.56 054 

8.56 083 

11.43 917 

9.99 

971 

55 


6 

8.56 400 

8.56 429 

11.43 571 

9.99 

971 

54 


7 

8.56 743 

8.56 773 

11.43 227 

9.99 

970 

53 


8 

8.57 084 

8.57 114 

11.42 886 

9.99 

970 

52 


9 

8.57 421 

8.57 452 

11.42 548 

9.99 

969 

51 


10 

8.57 757 

8.57 788 

11.42 212 

9.99 

969 

50 


11 

8.58 089 

8.58 121 

11.41 879 

9.99 

968 

49 


12 

8.58 419 

8.58 451 

11.41 549 

9.99 

968 

48 


13 

8.58 747 

8.58 779 

11.41 221 

9.99 

967 

47 


14 

8.59 072 

8.59 105 

11.40 895 

9.99 

967 

46 


15 

8.59 395 

8.59 428 

11.40 572 

9.99 

967 

45 


16 

8.59 715 

8.59 749 

11.40.251 

9.99 

966 

44 

<N 

17 

8.60 033 

8.60 068 

11.39 932 

9.99 

966 

43 

0> 

18 

8.60 349 

8.60 384 

11.39 616 

9.99 

965 

42 

bfi 

c3 

19 

8.60 662 

8.60 698 

11.39 302 

9.99 

964 

41 

a 

20 

8.60 973 

8.61 009 

11.38 991 

9.99 

964 

40 

3 m 

21 

8.61 282 

8.61 319 

11.38 681 

9.99 

963 

39 

3 9 

22 

8.61 589 

8.61 626 

11.38 374 

9.99 

963 

38 

Ah 

23 

8.61 894 

8.61 931 

11.38 069 

9.99 

962 

37 

$1 

24 

8.62 196 

8.62 234 

11.37 766 

9.99 

962 

36 

M bO 

o 

25 

8.62 497 

8.62 535 

11.37 465 

9.99 

961 

35 

rO C4H 

26 

8.62 795 

8.62 834 

11.37 166 

9.99 

961 

34 

> ° 

27 

8.63 091 

8.63 131 

11.36 869 

9.99 

960 

33 

* g 

28 

8.63 385 

8.63 426 

11.36 574 

9.99 

960 

32 

°J3 

29 

8.63 678 

8.63 718 

11.36 282 

9.99 

959 

31 

> > 

30 

8.63 968 

8.64 009 

11.35 991 

9.99 

959 

30 


31 

8.64 256 

8.64 298 

11.35 702 

9.99 

958 

29 

-Q-g 

32 

8.64 543 

8.64 585 

11.35 415 

9.99 

958 

28 

’ rj 

33 

8.64 827 

8.64 870 

11.35 130 

9.99 

957 

27 


34 

8.65 110 

8.65 154 

11.34 846 

9.99 

956 

26 

ra o3 

35 

8.65 391 

8.65 435 

11.34 565 

9.99 

956 

25 


36 

8.65 670 

8.65 715 

11.34 285 

9.99 

955 

24 

.2 o 

-P *H 

37 

8.65 947 

8.65 993 

11.34 007 

9.99 

955 

23 


38 

8.66 223 

8.66 269 

11.33 731 

9.99 

954 

22 

■3® 

Q,’ -1 

39 

8.66 497 

8.66 543 

11.33 457 

9.99 

954 

21 

' Eh 

0) O 

40 

8.66 769 

8.66 816 

11.33 184 

9.99 

953 

20 

-e as 
.2 1-1 

41 

8.67 039 

8.67 087 

11.32 913 

9.99 

952 

19 


42 

8.67 308 

8.67 356 

11.32 644 

9.99 

952 

18 

Is 0 

43 

8.67 575 

8.67 624 

11.32 376 

9.99 

951 

17 


44 

8.67 841 

8.67 890 

11.32 110 

9.99 

951 

16 

ts 

45 

8.68 104 

8.68 154 

11.31 846 

9.99 

950 

15 


46 

8.68 367 

8.68 417 

11.31 583 

9.99 

949 

14 


47 

8.68 627 

8.68 678 

11.31 322 

9.99 

949 

13 


48 

8.68 886 

8.68 938 

11.31 062 

9.99 

948 

12 


49 

8.69 144 

8.69 196 

11.30 804 

9.99 

948 

11 


50 

8.69 400 

8.69 453 

11.30 547 

9.99 

947 

10 


51 

8.69 654 

8.69 708 

11.30 292 

9.99 

946 

9 


52 

8.69 907 

8.69 962 

11.30 038 

9.99 

946 

8 


53 

8.70 159 

8.70 214 

11.29 786 

9.99 

945 

7 


54 

8.70 409 

8.70 465 

11.29 535 

9.99 

944 

6 


55 

8.70 658 

8.70 714 

11.29 286 

9.99 

944 

5 


56 

8.70 905 

8.70 962 

11.29 038 

9.99 

943 

4 


57 

8.71 151 

8.71 208 

11.28 792 

9.99 

942 

3 


58 

8.71 395 

8.71 453 

11.28 547 

9.99 

942 

2 


59 

8 71 638 

8.71 697 

11.28 303 

9.99 

941 

1 


60 

8.71 880 

8.71 940 

11.28 060 

9.99 

940 

0 



L Cos 

L Cot 

L Tan 

L Sin 

' 

Prop. Pts. 


87 ' 














































28 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


3° 


' 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 


Prop. Pts. 

0 

8.71 880 


8.71 940 


11.28 060 

9.99 940 

60 


1 

8.72 120 

240 

8.72 181 

241 

11.27 819 

9.99 940 

59 


2 

8.72 359 

239 

8.72 420 

239 

11.27 580 

9.99 939 

58 


3 

8.72.597 

238 

8.72 659 

239 

11.27 341 

9.99 938 

57 


4 

8.72 834 

237 

8.72 896 

237 

11.27 104 

9.99 938 

56 


5 

8.73 069 

235 

8.73 132 


11.26 868 

9.99 937 

55 


6 

8.73 303 

234 

8.73 366 

234 

11.26 634 

9.99 936 

54 


7 

8.73 535 

232 

8.73 600 

234 

11.26 400 

9.99 936 

53 


8 

8.73 767 

232 

8.73 832 

232 

11.26 168 

9.99 935 

52 


9 

8.73 997' 

230 

8.74 063 

231 

229 

11.25 937 

9.99 934 

51 


10 

8.74 226 


8.74 292 

11.25 708 

9.99 934 

50 


11 

8.74 454 

228 

8.74 521 

229 

11.25 479 

9.99 933 

49 


12 

8.74 680 

226 

8.74 748 

227 

11.25 252 

9.99 932 

48 


13 

8.74 906 

226 

8.74 974 

226 

11.25 026 

9.99 932 

47 


14 

8.75 130 

224 

8.75 199 

225 

11.24 801 

9.99 931 

46 


15 

8.75 353 


8.75 423 


11.24 577 

9.99 930 

45 


16 

8.75 575 

222 

8.75 645 

222 

11.24 355 

9.99 929 

44 


17 

8.75 795 

220 

8.75 867 

222 

11.24 133 

9.99 929 

43 


18 

8.76 015 

220 

8.76 087 

220 

11.23 913 

9.99 928 

42 


19 

8.76 234 

219 

8.76 306 

219 

219 

11.23 694 

9.99 927 

41 


20 

8.76 451 


8.76 525 

11.23 475 

9.99 926 

40 


21 

8.76 667 

216 

8.76 742 

217 

11.23 258 

9.99 926 

39 

| 

22 

8.76 883 

216 

8.76 958 

216 

11.23 042 

9.99 925 

38 

c3 

23 

8.77 097 

214 

8.77 173 

215 

11.22 827 

9.99 924 

37 

CM 

24 

8.77 310 

213 

oio 

8.77 387 

214 

213 

11.22 613 

9.99 923 

36 


25 

8.77 522 

ZlZ 

8.77 600 

11.22 400 

9.99 923 

35 

o 

26 

8.77 733 

211 

8.77 811 

211 

11.22 189 

9.99 922 

31 

t 

27 

8.77 943 

210 

8.78 022 

211 

11.21 978 

9.99 921 

33 

o 

28 

8.78 152 

209 

8.78 232 

210 

11.21 768 

9.99 920 

32 

o 

29 

8.78 360 

208 

8.78 441 

209 

OAC 

11.21 559 

9.99 920 

31 

& 

30 

8.78 568 

208 

8.78 649 


11.21 351 

9.99 919 

30 

Fh 

o 

31 

8.78 774 

206 

8.78 855 

206 

11.21 145 

9.99 918 

29 


32 

8.78 979 

205 

8.79 061 

206 

11.20 939 

9.99 917 

28 

a> 

M 

33 

8.79 183 

204 

8.79 266 

205 

11.20 734 

9.99 917 

27 

c3 

34 

8.79 386 

203 

8.79 470 

204 

11.20 530 

9.99 916 

26 

© 

35 

8.79 588 

202 

8.79 673 

4\JO 

11.20 327 

9.99 915 

25 


36 

8.79 789 

201 

8.79 875 

202 

11.20 125 

9.99 914 

24 

o 

37 

8.79 990 

201 

8.80 076 

201 

11.19 924 

9.99 913 

23 

a 

38 

8.80 189 

199 

8.80 277 

201 

11.19 723 

9.99 913 

22 

o 

39 

8.80 388 

199 

197 

8.80 476 

199 

1 QQ 

11.19 524 

9.99 912 

21 

© 

© 

40 

8.80 585 

8.80 674 

jl yo 

11.19 326 

9.99 911 

20 

CQ 

41 

8.80 782 

197 

8.80 872 

198 

11.19 128 

9.99 910 

19 


42 

8.80 978 

196 

8.81 068 

196 

11.18 932 

9.99 909 

18 


43 

8.81 173 

195 

8.81 264 

196 

11.18 736 

9.99 909 

17 


44 

8.81 367 

194 

lOQ 

8.81 459 

195 

11.18 541 

9.99 908 

16 


45 

8.81 560 

lyo 

8.81 653 


11.18 347 

9.99 907 

15 


46 

8.81 752 

192 

8.81 846 

193 

11.18 154 

9.99 906 

14 


47 

8.81 944 

192 

8.82 038 

192 

11.17 962 

9.99 905 

13 


48 

8.82 134 

190 

8.82 230 

192 

11.17 770 

9.99 904 

12 


49 

8.82 324 

190 

8.82 420 

190 
i on 

11.17 580 

9.99 904 

11 


50 

8.82 513 

189 

8.82 610 


11.17 390 

9.99 903 

10 


51 

8.82 701 

188 

8.82 799 

189 

11.17 201 

9.99 902 

9 


52 

8.82 888 

187 

8.82 987 

188 

11.17 013 

9.99 901 

8 


53 

8.83 075 

187 

8.83 175 

188 

11.16 825 

9.99 900 

7 


54 

8.83 261 

186 

185 

8.83 361 

186 

11.16 639 

9.99 899 

6 


55 

8.83 446 

8.83 547 

186 

11.16 453 

9.99 898 

5 


56 

8.83 630 

184 

8.83 732 

185 

11.16 268 

9.99 898 

4 


57 

8.83 813 

183 

8.83 916 

184 

11.16 084 

9.99 897 

3 


58 

8.83 996 

183 

8.84 100 

184 

11.15 900 

9.99 896 

2 


59 

8.84 177 

181 

181 

8.84 282 

182 

11.15 718 

9.99 895 

1 


60 

8.84 358 

8.84 464 

182 

11.15 536 

9.99 894 

0 



L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


86 ‘ 




























































FIVE-PLACE LOGARITHMS OF FUNCTIONS 


Proportional Parts for 3° 


" 

241 

240 

239 

238 

237 

236 

235 

234 

232 

231 

230 

6 

24.1 

24.0 

23.9 

23.8 

23.7 

23.6 

23.5 

23.4 

23.2 

23.1 

23.0 

7 

28.1 

28.0 

27.9 

27.8 

27.6 

27.5 

27.4 

27.3 

27.1 

27.0 

26.8 

8 

32.1 

32.0 

31.9 

31.7 

31.6 

31.5 

31.3 

31.2 

30.9 

30.8 

30.7 

9 

36.2 

36.0 

35.8 

35.7 

35.6 

35.4 

35.2 

35.1 

34.8 

34.7 

34.5 

10 

40.2 

40.0 

39.8 

39.7 

39.5 

39.3 

39.2 

39.0 

38.7 

38.5 

38.3 

20 

80.3 

80.0 

79.7 

79.3 

79.0 

78.7 

78.3 

78.0 

77.3 

77.0 

76.7 

30 

120.5 

120.0 

119.5 

119.0 

118.5 

118.0 

117.5 

117.0 

116.0 

115.5 

115.0 

40 

160.7 

160.0 

159.3 

158.7 

158.0 

157.3 

156.7 

156.0 

154.7 

154.0 

153.3 

50 

200.8 

200.0 

199.2 

198.3 

197.5 

196.7 

195.8 

195.0 

193.3 

192.5 

191.7 

" 

229 

228 

227 

226 

225 

224 

223 

222 

220 

219 

217 

6 

22.9 

22.8 

22.7 

22.6 

22.5 

22.4 

22.3 

22.2 

22.0 

21.9 

21.7 

7 

26.7 

26.6 

26.5 

26.4 

26.2 

26.1 

26.0 

25.9 

25.7 

25.6 

25.3 

8 

30.5 

30.4 

30.3 

30.1 

30.0 

29.9 

29.7 

29.6 

29.3 

29.2 

28.9 

9 

34.4 

34.2 

34.0 

33.9 

33.8 

33.6 

33.4 

33.3 

33.0 

32.9 

32.6 

10 

38.2 

38.0 

37.8 

37.7 

37.5 

37.3 

37.2 

37.0 

36.7 

36.5 

36.2 

20 

76.3 

76.0 

75.7 

75.3 

75.0 

74.7 

74.3 

74.0 

73.3 

73.0 

72.3 

30 

114.5 

114.0 

113.5 

113.0 

112.5 

112.0 

111.5 

111.0 

110.0 

109.5 

108.5 

40 

152.7 

152.0 

151.3 

150.7 

150.0 

149.3 

148.7 

148.0 

146.7 

146.0 

144.7 

50 

190.8 

190.0 

189.2 

188.3 

187.5 

186.7 

185.8 

185.0 

183.3 

182.5 

180.8 

" 

216 

215 

214 

213 

212 

211 

210 

209 

208 

206 

205 

6 

21.6 

21.5 

21.4 

21.3 

21.2 

21.1 

21.0 

20.9 

20.8 

20.6 

20.5 

7 

25.2 

25.1 

25.0 

24.9 

24.7 

24.6 

24.5 

24.4 

24.3 

24.0 

23.9 

8 

28.8 

28.7 

28.5 

28.4 

28.3 

28.1 

28.0 

27.9 

27.7 

27.5 

27.3 

9 

32.4 

32.2 

32.1 

32.0 

31.8 

31.6 

31.5 

31.4 

31.2 

30.9 

30.8 

10 

36.0 

35.8 

35.7 

35.5 

35.3 

35.2 

35.0 

34.8 

34.7 

34.3 

34.2 

20 

72.0 

71.7 

71.3 

71.0 

70.7 

70.3 

70.0 

69.7 

69.3 

68.7 

68.3 

30 

108.0 

107.5 

107.0 

106.5 

106.0 

105.5 

105.0 

104.5 

104.0 

103.0 

102.5 

40 

144.0 

143.3 

142.7 

142.0 

141.3 

140.7 

140.0 

139.3 

138.7 

137.3 

136.7 

50 

180.0 

179.2 

178.3 

177.5 

176.7 

175.8 

175.0 

174.2 

173.3 

171.7 

170.8 

" 

204 

203 

202 

201 

199 

198 

197 

196 

195 

194 

193 

6 

20.4 

20.3 

20.2 

20.1 

19.9 

19.8 

19.7 

19.6 

19.5 

19.4 

19.3 

7 

23.8 

23.7 

23.6 

23.4 

23.2 

23.1 

23.0 

22.9 

22.8 

22.6 

22.5 

8 

27.2 

27.1 

26.9 

26.8 

26.5 

26.4 

26.3 

26.1 

26.0 

25.9 

25.7 

9 

30.6 

30.4 

30.3 

30.2 

29.8 

29.7 

29.6 

29.4 

29.2 

29.1 

29.0 

10 

34.0 

33.8 

33.7 

33.5 

33.2 

33.0 

32.8 

32.7 

32.5 

32.3 

32.2 

20 

68.0 

67.7 

67.3 

67.0 

66.3 

66.0 

65.7 

65.3 

65.0 

64.7 

64.3 

30 

102.0 

101.5 

101.0 

100.5 

99.5 

99.0 

98.5 

98.0 

97.5 

97.0 

96.5 

40 

136.0 

135.3 

134.7 

134.0 

132.7 

132.0 

131.3 

130.7 

130.0 

129.3 

128.7 

50 

170.0 

169.2 

168.3 

167.5 

165.8 

165.0 

164.2 

163.3 

162.5 

161.7 

160.8 

« 

192 

190 

189 

188 

187 

186 

185 

184 

183 

182 

181 

6 

19.2 

19.0 

18.9 

18.8 

18.7 

18.6 

18.5 

18.4 

18.3 

18.2 

18.1 

7 

22.4 

22.2 

22.1 

21.9 

21.8 

21.7 

21.6 

21.5 

21.4 

21.2 

21.1 

8 

25.6 

25.3 

25.2 

25.1 

24.9 

24.8 

24.7 

24.5 

24.4 

24.3 

24.1 

9 

28.8 

28.5 

28.4 

28.2 

28.1 

27.9 

27.8 

27.6 

27.4 

27.3 

27.2 

10 

32.0 

31.7 

31.5 

31.3 

31.2 

31.0 

30.8 

30.7 

30.5 

30.3 

30.2 

20 

64.0 

63.3 

63.0 

62.7 

62.3 

62.0 

61.7 

61.3 

61.0 

60.7 

60.3 

30 

96.0 

95.0 

94.5 

94.0 

93.5 

93.0 

92.5 

92.0 

91.5 

91.0 

90.5 

40 

128.0 

126.7 

126.0 

125.3 

124.7 

124.0 

123.3 

122.7 

122.0 

121.3 

120.7 

50 

160.0 

158.3 

157.5 

156.7 

155.8 

155.0 

154.2 

153.3 

152.5 

151.7 

150.8 






























































30 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


4° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

0 

8.84 358 


8.84 464 


11.15 536 

9.99 894 

60 


1 

8.84 539 

181 

8.84 646 

182 

11.15 354 

9.99 893 

59 


2 

8.84 718 

179 

8.84 826 

180 

11.15 174 

9.99 892 

58 


3 

8.84 897 

179 

8.85 006 

180 

11.14 994 

9.99 891 

57 


4 

8.85 075 

178 

8.85 185 

179 

11.14 815 

9.99 891 

56 


5 

8.85 252 


8.85 363 


11.14 637 

9.99 890 

55 


6 

8.85 429 

177 

8.85 540 

177 

11.14 460 

9.99 889 

54 


7 

8.85 605 

176 

8.85 717 

177 

11.14 283 

9.99 888 

53 


8 

8.85 780 

175 

8.85 893 

176 

11.14 107 

9.99 887 

52 


9 

8.85 955 

175 

8.86 069 

176 

11.13 931 

9.99 886 

51 


10 

8.86 128 


8.86 243 


11.13 757 

9.99 885 

50 


11 

8.86 301 

173 

8.86 417 

174 

11.13 583 

9.99 884 

49 


12 

8.86 474 

173 

8.86 591 

174 

11.13 409 

9.99 883 

48 


13 

8.86 645 

171 

8.86 763 

172 

11.13 237 

9.99 882 

47 


14 

8.86 816 

171 

8.86 935 

172 

11.13 065 

9.99 881 

46 


15 

8.86 987 


8.87 106 


11.12 894 

9.99 880 

45 


16 

8.87 156 

169 

8.87 277 

171 

11.12 723 

9.99 879 

44 


17 

8.87 325 

169 

8.87 447 

170 

11.12 553 

9.99 879 

43 


18 

8.87 494 

169 

8.87 616 

169 

11.12 384 

9.99 878 

42 


19 

8.87 661 

167 

1C8 

8.87 785 

169 

11.12 215 

9.99 877 

41 


20 

8.87 829 


8.87 953 


11.12 047 

9.99 876 

40 


21 

8.87 995 

166 

8.88 120 

167 

11.11 880 

9.99 875 

39 

GO 

•43 

22 

8.88 161 

166 

8.88 287 

167 

11.11 713 

9.99 874 

38 

X, 

c3 

23 

8.88 326 

165 

8.88 453 

166 

11.11 547 

9.99 873 

37 

Ph 

24 

8.88 490 

164 

164 

8.88 618 

165 

165 

11.11 382 

9.99 872 

36 

"oS 

25 

8.88 654 


8.88 783 


11.11 217 

9.99 871 

35 

O 

_o 

26 

8.88 817 

163 

8.88 948 

165 

11.11 052 

9.99 870 

34 


27 

8.88 980 

163 

8.89 111 

163 

11.10 889 

9.99 869 

33 

O 

28 

8.89 142 

162 

8.89 274 

163 

11.10 726 

9.99 868 

32 

a 

o 

29 

8.89 304 

162 

8.89 437 

163 

161 

11.10 563 

9.99 867 

31 

>4 

P-i 

30 

8.89 464 

160 

8.89 598 


11.10 402 

9.99 866 

30 


31 

8.89 625 

161 

8.89 760 

162 

11.10 240 

9.99 865 

29 

o 

32 

8.89 784 

159 

8.89 920 

160 

11.10 080 

9.99 864 

28 

<D 

33 

8.89 943 

159 

8.90 080 

160 

11.09 920 

9.99 863 

27 


34 

8.90 102 

159 

8.90 240 

160 

11.09 760 

9.99 862 

26 

a 

35 

8.90 260 

lOo 

8.90 399 

±0*7 

11.09 601 

9.99 861 

25 


36 

8.90 417 

157 

8.90 557 

158 

11.09 443 

9.99 860 

24 

o 

37 

8.90 574 

157 

8.90 715 

158 

11.09 285 

9.99 859 

23 

a 

a 

38 

8.90 730 

156 

8.90 872 

157 

11.09 128 

9.99 858 

22 

o 

39 

8.90 885 

155 

8.91 029 

157 

156 

11.08 971 

9.99 857 

21 

0) 

3 

40 

8.91 040 

lOO 

8.91 185 

11.08 815 

9.99 856 

20 

CO 

41 

8.91 195 

155 

8.91 340 

155 

11.08 660 

9.99 855 

19 


42 

8.91 349 

154 

8.91 495 

155 

11.08 505 

9.99 854 

18 


43 

8.91 502 

153 

8.91 650 

155 

11.08 350 

9.99 853 

17 


44 

8.91 655 

153 

152 

8.91 803 

153 

154 

11.08 197 

9.99 852 

16 


45 

8.91 807 

8.91 957 

11.08 043 

9.99 851 

15 


46 

8.91 959 

152 

8.92 110 

153 

11.07 890 

9.99 850 

14 


47 

8.92 110 

151 

8.92 262 

152 

11.07 738 

9.99 848 

13 


48 

8.92 261 

151 

8.92 414 

152 

11.07 586 

9.99 847 

12 


49 

8.92 411 

150 

150 

8.92 565 

151 

11.07 435 

9.99 846 

11 


50 

8.92 561 

8.92 716 

101 

11.07 284 

9.99 845 

10 


51 

8.92 710 

149 

8.92 866 

150 

11.07 134 

9.99 844 

9 


52 

8.92 859 

149 

8.93 016 

150 

11.06 984 

9.99 843 

8 


53 

8.93 007 

148 

8.93 165 

149 

11.06 835 

9.99 842 

7 


54 

8.93 154 

147 

147 

8.93 313 

148 

149 

11.06 687 

9.99 841 

6 


55 

8.93 301 

8.93 462 


11.06 538 

9.99 840 

5 


56 

8.93 448 

147 

8.93 609 

147 

11.06 391 

9.99 839 

4 


57 

8.93 594 

146 

8.93 756 

147 

11.06 244 

9.99 838 

3 


58 

8.93 740 

146 

8.93 903 

147 

11.06 097 

9.99 837 

2 


59 

8.93 885 

145 

8.94 049 

146 

146 

11.05 951 

9.99 836 

1 


60 

8.94 030 

145 

8.94 195 

11.05 805 

9.99 834 

0 



L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


85 ‘ 























































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


31 


Proportional Parts for 4° 


n 

182 

181 

180 

179 

178 

177 

176 

6 

18.2 

18.1 

18.0 

17.9 

17.8 

17.7 

17.6 

7 

21.2 

21.1 

21.0 

20.9 

20.8 

20.6 

20.5 

8 

24.3 

24.1 

24.0 

23.9 

23.7 

23.6 

23.5 

9 

27.3 

27.2 

27.0 

26.8 

26.7 

26.6 

26.4 

10 

30.3 

30.2 

30.0 

29.8 

29.7 

29.5 

29.3 

20 

60.7 

60.3 

60.0 

59.7 

59.3 

59.0 

58.7 

30 

91.0 

90.5 

90.0 

89.5 

89.0 

88.5 

88.0 

40 

121.3 

120.7 

120.0 

119.3 

118.7 

118.0 

117.3 

50 

151.7 

150.8 

150.0 

149.2 

148.3 

147.5 

146.7 

// 

175 

174 

173 

172 

171 

170 

169 

6 

17.5 

17.4 

17.3 

17.2 

17.1 

17.0 

16.9 

7 

20.4 

20.3 

20.2 

20.1 

20.0 

19.8 

19.7 

8 

23.3 

23.2 

23.1 

22.9 

22.8 

22.7 

22.5 

9 

26.2 

26.1 

26.0 

25.8 

25.6 

25.5 

25.4 

10 

29.2 

29.0 

28.8 

28.7 

28.5 

28.3 

28.2 

20 

58.3 

58.0 

57.7 

57.3 

57.0 

56.7 

56.3 

30 

87.5 

87.0 

86.5 

86.0 

85.5 

85.0 

84.5 

40 

116.7 

116.0 

115.3 

114.7 

114.0 

113.3 

112.7 

50 

145.8 

145.0 

144.2 

143.3 

142.5 

141.7 

140.8 

// 

168 

167 

166 

165 

164 

163 

162 

6 

16.8 

16.7 

16.6 

16.5 

16.4 

16.3 

16.2 

7 

19.6 

19.5 

19.4 

19.2 

19.1 

19.0 

18.9 

8 

22.4 

22.3 

22.1 

22.0 

21.9 

21.7 

21.6 

9 

25.2 

25.0 

24.9 

24.8 

24.6 

24.4 

24.3 

10 

28.0 

27.8 

27.7 

27.5 

27.3 

27.2 

27.0 

20 

56.0 

55.7 

55.3 

55.0 

54.7 

54.3 

54.0 

30 

84.0 

83.5 

83.0 

82.5 

82.0 

81.5 

81.0 

40 

112.0 

111.3 

110.7 

110.0 

109.3 

108.7 

108.0 

50 

140.0 

139.2 

138.3 

137.5 

136.7 

135.8 

135.0 


161 

160 

159 

158 

157 

156 

155 

6 

16.1 

16.0 

15.9 

15.8 

15.7 

15.6 

15.5 

7 

18.8 

18.7 

18.6 

18.4 

18.3 

18.2 

18.1 

8 

21.5 

21.3 

21.2 

21.1 

20.9 

20.8 

20.7 

9 

24.2 

24.0 

23.8 

23.7 

23.6 

23.4 

23.2 

10 

26.8 

26.7 

26.5 

26.3 

26.2 

26.0 

25.8 

20 

53.7 

53.3 

53.0 

52.7 

52.3 

52.0 

51.7 

30 

80.5 

80.0 

79.5 

79.0 

78.5 

78.0 

77.5 

40 

107.3 

106.7 

106.0 

105.3 

104.7 

104.0 

103.3 

50 

134.2 

133.3 

132.5 

131.7 

130.8 

130.0 

129.2 

// 

154 

153 

152 

151 

150 

149 

148 

g 

15.4 

15.3 

15.2 

15.1 

15.0 

14.9 

14.8 

7 

18.0 

17.8 

17.7 

17.6 

17.5 

17.4 

17.3 

g 

20 5 

20.4 

20.3 

20.1 

20.0 

19.9 

19.7 

9 

23.1 

23.0 

22.8 

22.6 

22.5 

22.4 

22.2 

10 

25.7 

25.5 

25.3 

25.2 

25.0 

24.8 

24.7 

20 

51.3 

51.0 

50.7 

50.3 

50.0 

49.7 

49.3 

30 

77 0 

76.5 

76.0 

75.5 

75.0 

74.5 

74.0 

40 

102.7 

102.0 

101.3 

100.7 

100.0 

99.3 

98.7 

50 

128.3 

127.5 

126.7 

125.8 

125.0 

124.2 

123.3 


For 147, 146, and 145 see page 32. 







































32 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


5 ° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

0 

1 

2 

3 

4 

8.94 030 
8.94 174 
8.94 317 
8.94 461 
8.94 603 

144 

143 

144 

142 

143 

141 

142 
141 
140 
140 

139 

139 

139 

138 

138 

137 

137 

136 

136 

136 

135 

135 

134 

134 

133 

133 

133 

132 

132 

131 

131 

131 

130 

130 

129 

129 

129 

128 

128 

128 

127 

127 

126 

126 

126 

125 

125 

124 

125 

123 

124 
123 
123 
122 
122 

122 

121 

121 

121 

120 

8.94 195 
8.94 340 
8.94 485 
8.94 630 
8.94 773 

145 

145 

145 

143 

144 

143 

142 

142 

142 

141 

140 

141 

139 

140 

138 

139 
138 

137 

138 

136 

137 

135 

136 
135 
135 

134 

134 

133 

133 

133 

132 

132 

131 

131 

131 

130 

130 

130 

129 

128 

129 

128 

127 

128 
127 

126 

126 

126 

126 

125 

125 

124 

124 

124 

123 

123 

123 

122 

122 

122 

11.05 805 
11.05 660 
11.05 515 
11.05 370 
11.05 227 

9.99 834 
9.99 833 
9.99 832 
9.99 831 
9.99 830 

60 

59 

58 

57 

56 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

6 

7 

8 

9 

10 

20 

30 

40 

50 

147 

14.7 

17.2 

19.6 
22.0 

24.5 
49.0 

73.5 
98.0 

122.5 

143 

14.3 

16.7 

19.1 

21.4 

23.8 

47.7 

71.5 
95.3 

119.2 

139 

13.9 

16.2 

18.5 

20.9 

23.2 

46.3 

69.5 

92.7 

115.8 

135 

13.5 

15.8 
18.0 
20.2 

22.5 
45.0 

67.5 
90.0 

112.5 

131 

13.1 

15.3 

17.5 

19.6 

21.8 

43.7 

65.5 

87.3 

109.2 

127 

12.7 

14.8 

16.9 
19.0 

21.2 

42.3 

63.5 
84.7 

105.8 

123 

12.3 

14.4 

16.4 

18.4 

20.5 
41.0 

61.5 
82.0 

102.5 

146 

14.6 
17.0 

19.5 

21.9 

24.3 

48.7 
73.0 

97.3 

121.7 

142 

14.2 

16.6 

18.9 

21.3 

23.7 

47.3 
71.0 

94.7 

118.3 

138 

13.8 
16.1 

18.4 
20.7 
23.0 
46.0 
69.0 
92.0 

[115.0 

134 

13.4 

15.6 

17.9 
20.1 

22.3 

44.7 
67.0 

89.3 

111.7 

130 

13.0 

15.2 

17.3 

19.5 

21.7 

43.3 
65.0 

86.7 

108.3 

126 

12.6 

14.7 

16.8 

18.9 
21.0 
42.0 
63.0 
84.0 

105.0 

122 

12.2 

14.2 

16.3 

18.3 

20.3 
40.7 
61.0 

81.3 

101.7 

145 

14.5 

16.9 
19.3 
21.8 

24.2 

48.3 

72.5 

96.7 
120.8 

141 

14.1 

16.4 

18.8 

21.2 

23.5 
47.0 

70.5 
94.0 

117.5 

137 

13.7 
16.0 

18.3 

20.6 

22.8 

45.7 

68.5 

91.3 

114.2 

133 

13.3 

15.5 

17.7 
20.0 
22.2 

44.3 

66.5 

88.7 
110.8 

129 

12.9 
15.0 

17.2 

19.4 

21.5 
43.0 

64.5 
86.0 

107.5 

125 

12.5 

14.6 

16.7 

18.8 

20.8 

41.7 

62.5 

83.3 

104.2 

121 

12.1 

14.1 

16.1 
18.2 
20.2 

40.3 

60.5 

80.7 
100.8 

144 

14.4 

16.8 

19.2 
21.6 
24.0 
48.0 
72.0 
96.0 

120.0 

140 

14.0 

16.3 

18.7 
21.0 

23.3 

46.7 
70.0 

93.3 

116.7 

136 

13.6 

15.9 
18.1 

20.4 

22.7 

45.3 
68.0 

90.7 

113.3 

132 

13.2 

15.4 

17.6 

19.8 
22.0 
44.0 
66.0 
88.0 

110.0 

128 

12.8 

14.9 

17.1 

19.2 

21.3 

42.7 
64.0 

85.3 

106.7 

124 

12.4 

14.5 

16.5 

18.6 

20.7 
41.3 
62.0 

82.7 

103.3 

120 

12.0 

14.0 

16.0 

18.0 

20.0 

40.0 

60.0 

80.0 

100.0 

5 

6 

7 

8 

9 

8.94 746 

8.94 887 

8.95 029 
8.95 170 
8.95 310 

8.94 917 

8.95 060 
8.95 202 
8.95 344 
8.95 486 

11.05 083 
11.04 940 
11.04 798 
11.04 656 
11.04 514 

9.99 829 
9.99 828 
9.99 827 
9.99 825 
9.99 824 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

8.95 450 
8.95 589 
8.95 728 

8.95 867 

8.96 005 

8.95 627 
8.95 767 

8.95 908 

8.96 047 
8.96 187 

11.04 373 
11.04 233 
11.04 092 
11.03 953 
11.03 813 

9.99 823 
9.99 822 
9.99 821 
9.99 820 
9.99 819 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

8.96 143 
8.96 280 
8.96 417 
8.96 553 
8.96 689 

8.96 325 
8.96 464 
8.96 602 
8.96 739 
8.96 877 

11.03 675 
11.03 536 
11.03 398 
11.03 261 
11.03 123 

9.99 817 
9.99 816 
9.99 815 
9.99 814 
9.99 813 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

8.96 825 

8.96 960 

8.97 095 
8.97 229 
8.97 363 

8.97 013 
8.97 150 
8.97 285 
8.97 421 
8.97 556 

11.02 987 
11.02 850 
11.02 715 
11.02 579 
11.02 444 

9.99 812 
9.99 810 
9.99 809 
9.99 808 
9.99 807 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

8.97 496 
8.97 629 
8.97 762 

8.97 894 

8.98 026 

8.97 691 
8.97 825 

8.97 959 

8.98 092 
8.98 225 

11.02 309 
11.02 175 
11.02 041 
11.01 908 
11.01 775 

9.99 806 
9.99 804 
9.99 803 
9.99 802 
9.99 801 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

8.98 157 
8.98 288 
8.98 419 
8.98 549 
8.98 679 

8.98 358 
8.98 490 
8.98 622 
8.98 753 
8.98 884 

11.01 642 
11.01 510 
11.01 378 
11.01 247 
11.01 116 

9.99 800 
9.99 798 
9.99 797 
9.99 796 
9.99 795 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

8.98 808 

8.98 937 

8.99 066 
8.99 194 
8.99 322 

8.99 015 
8.99 145 
8.99 275 
8.99 405 
8.99 534 

11.00 985 
11.00 855 
11.00 725 
11.00 595 
11.00 466 

9.99 793 
9.99 792 
9.99 791 
9.99 790 
9.99 788 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

8.99 450 
8.99 577 
8.99 704 
8.99 830 
8.99 956 

8.99 662 
8.99 791 
8.99 919 
9.00 046 
9.00 174 

11.00 338 
11.00 209 
11.00 081 
10.99 954 
10.99 826 

9.99 787 
9.99 786 
9.99 785 
9.99 783 
9.99 782 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.00 082 
9.00 207 
9.00 332 
9.00 456 
9.00 581 

9.00 301 
9.00 427 
9.00 553 
9.00 679 
9.00 805 

10.99 699 
10.99 573 
10.99 447 
10.99 321 
10.99 195 

9.99 781 
9.99 780 
9.99 778 
9.99 777 
9.99 776 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.00 704 
9.00 828 
9.00 951 
9.01 074 
9.01 196 

9.00 930 
9.01 055 
9.01 179 
9.01 303 
9.01 427 

10.99 070 
10.98 945 
10.98 821 
10.98 697 
10.98 573 

9.99 775 
9.99 773 
9.99 772 
9.99 771 
9.99 769 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.01 318 
9.01 440 
9.01 561 
9.01 682 
9.01 803 

9.01 550 
9.01 673 
9.01 796 
9.01 918 
9.02 040 

10.98 450 
10.98 327 
10.98 204 
10.98 082 
10.97 960 

9.99 768 
9.99 767 
9.99 765 
9.99 764 
9.99 763 

5 

4 

3 

2 

1 

60 

9.01 923 

9.02 162 

10.97 838 

9.99 761 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


84 ' 

























































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


33 


6 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


0 

9.01 923 


9.02 162 


10.97 838 

9.99 761 

60 

1 

9.02 043 

120 

9.02 283 

121 

10.97 717 

9.99 760 

59 

2 

9.02 163 

120 

9.02 404 

121 

10.97 596 

9.99 759 

58 

3 

9.02 283 

120 

9.02 525 

121 

10.97 475 

9.99 757 

57 

4 

9.02 402 

119 

118 

9.02 645 

120 

121 

10.97 355 

9.99 756 

56 

5 

9.02 520 

9.02 766 

10.97 234 

9.99 755 

55 

6 

9.02 639 

119 

9.02 885 

119 

10.97 115 

9.99 753 

54 

7 

9.02 757 

118 

9.03 005 

120 

10.96 995 

9.99 752 

53 

8 

9.02 874 

117 

9.03 124 

119 

10.96 876 

9.99 751 

52 

9 

9.02 992 

118 

117 

9.03 242 

118 

119 

10.96 758 

9.99 749 

51 

10 

9.03 109 

9.03 361 

10.96 639 

9.99 748 

50 

11 

9.03 226 

117 

9.03 479 

118 

10.96 521 

9.99 747 

49 

12 

9.03 342 

116 

9.03 597 

118 

10.96 403 

9.99 745 

48 

13 

9.03 458 

116 

9.03 714 

117 

10.96 286 

9.99 744 

47 

14 

9.03 574 

116 

116 

9.03 832 

118 

116 

10.96 168 

9.99 742 

46 

15 

9.03 690 

9.03 948 

10.96 052 

9.99 741 

45 

16 

9.03 805 

115 

9.04 065 

117 

10.95 935 

9.99 740 

44 

17 

9.03 920 

115 

9.04 181 

116 

10.95 819 

9.99 738 

43 

18 

9.04 034 

114 

9.04 297 

116 

10.95 703 

9.99 737 

42 

19 

9.04 149 

115 

113 

9.04 413 

116 

115 

10.95 587 

9.99 736 

41 

20 

9.04 262 

9.04 528 

10.95 472 

9.99 734 

40 

21 

9.04 376 

114 

9.04 643 

115 

10.95 357 

9.99 733 

39 

22 

9.04 490 

114 

9.04 758 

115 

10.95 242 

9.99 731 

38 

23 

9.04 603 

113 

9.04 873 

115 

10.95 127 

9.99 730 

37 

24 

9.04 715 

112 

113 

9.04 987 

114 

114 

10.95 013 

9.99 728 

36 

25 

9.04 828 

9.05 101 

10.94 899 

9.99 727 

35 

26 

9.04 940 

112 

9.05 214 

113 

10.94 786 

9.99 726 

34 

27 

9.05 052 

112 

9.05 328 

114 

10.94 672 

9.99 724 

33 

28 

9.05 164 

112 

9.05 441 

113 

10.94 559 

9.99 723 

32 

29 

9.05 275 

111 

111 

9.05 553 

112 

113 

10.94 447 

9.99 721 

31 

30 

9.05 386 

9.05 666 

10.94 334 

9.99 720 

30 

31 

9.05 497 

111 

9.05 778 

112 

10.94 222 

9.99 718 

29 

32 

9.05 607 

110 

9.05 890 

112 

10.94 110 

9.99 717 

28 

33 

9.05 717 

110 

9.06 002 

112 

10.93 998 

9.99 716 

27 

34 

9.05 827 

110 

110 

9.06 113 

111 

111 

10.93 887 

9.99 714 

26 

35 

9.05 937 

9.06 224 

10.93 776 

9.99 713 

25 

36 

9.06 046 

109 

9.06 335 

111 

10.93 665 

9.99 711 

24 

37 

9.06 155 

109 

9.06 445 

110 

10.93 555 

9.99 710 

23 

38 

9.06 264 

109 

9.06 556 

111 

10.93 444 

9.99 708 

22 

39 

9.06 372 

108 
i no 

9.06 666 

110 

109 

10.93 334 

9.99 707 

21 

40 

9.06 481 

iu y 

9.06 775 

10.93 225 

9.99 705 

20 

41 

9.06 589 

108 

9.06 885 

110 

10.93 115 

9.99 704 

19 

42 

9.06 696 

107 

9.06 994 

109 

10.93 006 

9.99 702 

18 

43 

9.06 804 

108 

9.07 103 

109 

10.92 897 

9.99 701 

17 

44 

9.06 911 

107 

9.07 211 

108 

109 

10.92 789 

9.99 699 

16 

45 

9.07 018 

1U t 

9.07 320 

10.92 680 

9.99 698 

15 

46 

9.07 124 

106 

9.07 428 

108 

10.92 572 

9.99 696 

14 

47 

9.07 231 

107 

9.07 536 

108 

10.92 464 

9.99 695 

13 

48 

9.07 337 

106 

9.07 643 

107 

10.92 357 

9.99 693 

12 

49 

9.07 442 

105 

9.07 751 

108 

1 07 

10.92 249 

9.99 692 

11 

50 

9.07 548 

IUD 

9.07 858 

io < 

10.92 142 

9.99 690 

10 

51 

9.07 653 

105 

9.07 964 

106 

10.92 036 

9.99 689 

9 

52 

9.07 758 

105 

9.08 071 

107 

10.91 929 

9.99 687 

8 

53 

9.07 863 

105 

9.08 177 

106 

10.91 823 

9.99 686 

7 

54 

9.07 968 

105 

9.08 283 

106 
l nfi 

10.91 717 

9.99 684 

6 

55 

9.08 072 

1U4 

9.08 389 

iUO 

10.91 611 

9.99 683 

5 

56 

9.08 176 

104 

9.08 495 

106 

10.91 505 

9.99 681 

4 

57 

9.08 280 

104 

9.08 600 

105 

10.91 400 

9.99 680 

3 

58 

9.08 383 

103 

9.08 705 

105 

10.91 295 

9.99 678 

2 

59 

9.08 486 

103 

9.08 810 

105 

1 04 

10.91 190 

9.99 677 

1 

60 

9.08 589 

lUo 

9.08 914 

1 ui 

10.91 086 

9.99 675 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 


Prop. Pts. 


121 120 119 


12.1 

14.1 

16.1 
18.2 
20.2 
40.3 
60.5 
80.7 

100.8 


118 

11.8 

13.8 

15.7 

17.7 

19.7 

39.3 
59.0 

78.7 

98.3 


115 

11.5 

13.4 
15.3 

17.2 

19.2 

38.3 

57.5 

76.7 

95.8 


112 

11.2 

13.1 

14.9 

16.8 

18.7 

37.3 
56.0 

74.7 

93.3 


109 

10.9 

12.7 
14.5 

16.4 
18.2 
36.3 

54.5 

72.7 


106 

10.6 

12.4 

14.1 

15.9 

17.7 
35.3 
53.0 

70.7 


12.0 

14.0 

16.0 

18.0 

20.0 

40.0 

60.0 

80.0 

100.0 


117 

11.7 

13.6 

15.6 

17.6 

19.5 
39.0 

58.5 
78.0 

97.5 


114 

11.4 

13.3 

15.2 

17.1 

19.0 

38.0 

57.0 

76.0 

95.0 


111 

11.1 

13.0 

14.8 

16.6 

18.5 
37.0 

55.5 
74.0 

92.5 


108 

10.8 

12.6 

14.4 

16.2 

18.0 

36.0 

54.0 

72.0 

90.0 


105 

10.5 
12.2 
14.0 
15.8 

17.5 
35.0 

52.5 
70.0 

87.5 


11.9 

13.9 

15.9 

17.8 

19.8 
39.7 
59.5 
79.3 
99.2 


116 

11.6 

13.5 

15.5 
17.4 

19.3 

38.7 
58.0 

77.3 

96.7 


113 

11.3 

13.2 

15.1 
17.0 
18.8 
37.7 
56.5 

75.3 

94.2 


110 

11.0 

12.8 

14.7 
16.5 

18.3 

36.7 
55.0 

73.3 

91.7 


107 

10.7 

12.5 

14.3 
16.0 

17.8 
35.7 

53.5 

71.3 
89.2 


104 

10.4 

12.1 

13.9 

15.6 
17.3 

34.7 
52.0 


Prop. Pts. 


83 








































































34 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


7° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

0 

1 

9.08 589 
9.08 692 

103 

9.08 914 
9.09 019 

105 

10.91 086 
10.90 981 

9.99 675 
9.99 674 

60 

59 

" 

105 

104 

103 

2 

9.08 795 

103 

9.09 123 

104 

10.90 877 

9.99 672 

58 

6 

10.5 

i 10.4 

: 10.3 

3 

9.08 897 

102 

9.09 227 

104 

10.90 773 

9.99 670 

57 

7 

12.3 

12.1 

12.0 

4 

9.08 999 

102 

102 

9.09 330 

103 

10.90 670 

9.99 669 

56 

8 

9 

10 

14.0 

1 c o 

i 13.9 

1 C A 

13.7 

1 c i 

5 

9.09 101 

9.09 434 

103 

10.90 566 

9.99 667 

55 

ID.o 

17.5 

10.0 

17.3 

i 10.^1 

17.2 

6 

9.09 202 

101 

9.09 537 

10.90 463 

9.99 666 

54 

20 

35.0 

' 34.7 

34.3 

7 

9.09 304 

102 

9.09 640 

103 

10.90 360 

9.99 664 

53 

30 

52.5 

52.0 

' 51.5 

8 

9.09 405 

101 

9.09 742 

102 

10.90 258 

9.99 663 

52 

40 

70.0 

' 69.3 

68.7 

9 

9.09 506 

101 

100 

9.09 845 

103 

10.90 155 

9.99 661 

51 

50 

87.5 

86.7 

85.8 

10 

9.09 606 

9.09 947 

102 

10.90 053 

9.99 659 

50 



101 

100 

11 

9.09 707 

101 

9.10 049 

10.89 951 

9.99 658 

49 

" 

102 

12 

9.09 807 

100 

9.10 150 

101 

10.89 850 

9.99 656 

48 


10.2 
11 9 

10.1 
11 8 

10.0 

11 7 

13 

9.09 907 

100 

9.10 252 

102 

10.89 748 

9.99 655 

47 

Q 

7 

14 

9.10 006 

99 

100 

9.10 353 

101 

10.89 647 

9.99 653 

46 

8 

13^6 

13.5 

13.3 

15 

9.10 106 

9.10 454 

101 

10.89 546 

9.99 651 

45 

9 

15.3 

15.2 
16.8 
33 7 

15.0 

16.7 

33 3 

16 

9.10 205 

99 

9.10 555 

10.89 445 

9.99 650 

44 

10 

20 

17.0 
34 o 

17 

9.10 304 

99 

9.10 656 

101 

10.89 344 

9.99 648 

43 

30 

5i!o 

50^5 

50^0 

18 

9.10 402 

98 

9.10 756 

100 

10.89 244 

9.99 647 

42 

40 

68.0 

67.3 

66.7 

19 

9.10 501 

99 

98 

9.10 856 

100 

100 

10.89 144 

9.99 645 

41 

50 

85.0 

84.2 

83.3 

20 

9.10 599 

9.10 956 

10.89 044 

9.99 643 

40 





21 

9.10 697 

98 

9.11 056 

100 

10.88 944 

9.99 642 

39 

» 

99 

98 

97 

22 

9.10 795 

98 

9.11 155 

99 

10.88 845 

9.99 640 

38 




9.7 

23 

9.10 893 

98 

9.11 254 

99 

10.88 746 

9.99 638 

37 

6 

9.9 

9.8 

24 

9.10 990 

97 

97 

9.11 353 

99 

QQ 

10.88 647 

9.99 637 

36 

7 

8 

11.6 

13.2 

11.4 

13.1 

11.3 

12.9 

25 

9.11087 

9.11 452 

yy 

99 

10.88 548 

9.99 635 

35 

9 

14.8 

14.7 

14.6 

26 

9.11 184 

97 

9.11 551 

10.88 449 

9.99 633 

34 

10 

16.5 

16.3 

16.2 

27 

28 

9.11 281 
9.11 377 

97 

96 

9.11 649 
9.11 747 

98 

98 

10.88 351 
10.88 253 

9.99 632 
9.99 630 

33 

32 

20 

30 

40 

33.0 
49.5 
66 0 

32.7 
49.0 
65 3 

32.3 

48.5 

64 7 

29 

9.11 474 

97 

Q A 

9.11 845 

98 

QQ 

10.88 155 

9.99 629 

31 

50 

82.5 

81.7 

80.8 

30 

9.11 570 

»D 

9.11 943 

yo 

97 

10.88 057 

9.99 627 

30 





31 

9.11 666 

96 

9.12 040 

10.87 960 

9.99 625 

29 

„ 

96 

95 

94 

32 

9.11 761 

95 

9.12 138 

98 

10.87 862 

9.99 624 

28 


33 

9.11 857 

96 

9.12 235 

97 

10.87 765 

9.99 622 

27 

6 

9.6 

9.5 

9.4 

34 

9.11 952 

95 

95 

9.12 332 

97 

96 

10.87 668 

9.99 620 

26 

7 

8 

9 

11.2 

11.1 

12.7 

14.2 

11.0 

12.5 

14.1 

35 

9.12 047 

9.12 428 

10.87 572 

9.99 618 

25 

14^4 

36 

9.12 142 

95 

9.12 525 

97 

10.87 475 

9.99 617 

24 

10 

16.0 

15.8 

15.7 

37 

9.12 236 

94 

9.12 621 

96 

10.87 379 

9.99 615 

23 

20 

32.0 

31.7 

31.3 

38 

9.12 331 

95 

9.12 717 

96 

10.87 283 

9.99 613 

22 

30 

48.0 

47.5 

47.0 

39 

9.12 425 

94 

94 

9.12 813 

96 

QA 

10.87 187 

9.99 612 

21 

40 

50 

64.0 

80.0 

63.3 

79.2 

62.7 

78.3 

40 

9.12 519 

9.12 909 

y o 

95 

10.87 091 

9.99 610 

20 

41 

9.12 612 

93 

9.13 004 

10.86 996 

9.99 608 

19 




91 

42 

9.12 706 

94 

9.13 099 

95 

10.86 901 

9.99 607 

18 


93 

92 

43 

9.12 799 

93 

9.13 194 

95 

10.86 806 

9.99 605 

17 

g 

9 3 

9 2 

9 1 

44 

9.12 892 

93 

93 

9.13 289 

95 

95 

10.86 711 

9.99 603 

16 

7 

10.9 

10> 

10i6 

45 

9.12 985 

9.13 384 

10.86 616 

9.99 601 

15 

8 

12.4 

12.3 

12.1 

46 

9.13 078 

93 

9.13 478 

94 

10.86 522 

9.99 600 

14 

9 

10 

14.0 
15 5 

13.8 
15 3 

13.6 

15 2 

47 

9.13 171 

93 

9.13 573 

95 

10.86 427 

9.99 598 

13 

20 

3i!o 

30^7 

30^3 

48 

9.13 263 

92 

9.13 667 

94 

10.86 333 

9.99 596 

12 

30 

46.5 

46.0 

45.5 

49 

9.13 355 

92 

9.13 761 

94 

10.86 239 

9.99 595 

11 

40 

62.0 

61.3 

60.7 



92 

QQ 

50 

77.5 

76.7 

75.8 

50 

9.13 447 

9.13 854 

yo 

94 

10.86 146 

9.99 593 

10 

51 

9.13 539 

92 

9.13 948 

10.86 052 

9.99 591 

9 





52 

9.13 630 

91 

9.14 041 

93 

10.85 959 

9.99 589 

8 

" 

90 

2 

1 

53 

9.13 722 

92 

9.14 134 

93 

10.85 866 

9.99 588 

7 


9.0 

10.5 

0.2 

0.2 

0.1 

0.1 

54 

9.13 813 

91 

91 

9.14 227 

93 

93 

10.85 773 

9.99 586 

6 

6 

7 

55 

9.13 904 

9.14 320 

10.85 680 

9.99 584 

5 

8 

12.0 

0.3 

0.1 

56 

9.13 994 

90 

9.14 412 

92 

10.85 588 

9.99 582 

4 

9 

13.5 

0.3 

0.2 

57 

9.14 085 

91 

9.14 504 

92 

10.85 496 

9.99 581 

3 

10 

20 

30 

15.0 
30.0 
45 0 

0.3 

0.2 

58 

9.14 175 

90 

9.14 597 

93 

10.85 403 

9.99 579 

2 

0.7 

1 0 

0.3 

0 5 

59 

9.14 266 

91 

90 

9.14 688 

91 

92 

10.85 312 

9.99 577 

1 

40 

60'o 

1.3 

0.7 

60 

9.14 356 

9.14 780 

10.85 220 

9.99 575 

0 

50 

75.0 

1.7 

0.8 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


82 ' 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


35 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


0 

9.14 356 


9.14 780 


10.85 220 

9.99 575 

60 

1 

9.14 445 

89 

9.14 872 

92 

10.85 128 

9.99 574 

59 

2 

9.14 535 

90 

9.14 963 

91 

10.85 037 

9.99 572 

58 

3 

9.14 624 

89 

9.15 054 

91 

10.84 946 

9.99 570 

57 

4 

9.14 714 

90 

9.15 145 

91 

10.84 855 

9.99 568 

56 





91 




5 

9.14 803 


9.15 236 

10.84 764 

9.99 566 

55 

6 

9.14 891 

88 

9.15 327 

91 

10.84 673 

9.99 565 

54 

7 

9.14 980 

89 

9.15 417 

90 

10.84 583 

9.99 563 

53 

8 

9.15 069 

89 

9.15 508 

91 

10.84 492 

9.99 561 

52 

9 

9.15 157 

88 

88 

9.15 598 

90 

90 

10.84 402 

9.99 559 

51 

10 

9.15 245 

9.15 688 

10.84 312 

9.99 557 

50 

11 

9.15 333 

88 

9.15 777 

89 

10.84 223 

9.99 556 

49 

12 

9.15 421 

88 

9.15 867 

90 

10.84 133 

9.99 554 

48 

13 

9.15 508 

87 

9.15 956 

89 

10.84 044 

9.99 552 

47 

14 

9.15 596 

88 

9.16 046 

90 

10.83 954 

9.99 550 

46 









15 

9.15 683 


9.16 135 

oy 

10.83 865 

9.99 548 

45 

16 

9.15 770 

87 

9.16 224 

89 

10.83 776 

9.99 546 

44 

17 

9.15 857 

87 

9.16 312 

88 

10.83 688 

9.99 545 

43 

18 

9.15 944 

87 

9.16 401 

89 

10.83 599 

9.99 543 

42 

19 

9.16 030 

86 

9.16 489 

88 

10.83 511 

9.99 541 

41 





88 




20 

9.16 116 


9.16 577 

10.83 423 

9.99 539 

40 

21 

9.16 203 

87 

9.16 665 

88 

10.83 335 

9.99 537 

39 

22 

9.16 289 

86 

9.16 753 

88 

10.83 247 

9.99 535 

38 

23 

9.16 374 

85 

9.16 841 

88 

10.83 159 

9.99 533 

37 

24 

9.16 460 

86 

9.16 928 

87 

OQ 

10.83 072 

9.99 532 

36 

25 

9.16 545 


9.17 016 

OO 

10.82 984 

9.99 530 

35 

26 

9.16 631 

86 

9.17 103 

87 

10.82 897 

9.99 528 

34 

27 

9.16 716 

85 

9.17 190 

87 

10.82 810 

9.99 526 

33 

28 

9.16 801 

85 

9.17 277 

87 

10.82 723 

9.99 524 

32 

29 

9.16 886 

85 

9.17 363 

86 

87 

10.82 637 

9.99 522 

31 

30 

9.16 970 

O'! 

9.17 450 

10.82 550 

9.99 520 

30 

31 

9.17 055 

85 

9.17 536 

86 

10.82 464 

9.99 518 

29 

32 

9.17 139 

84 

9.17 622 

86 

10.82 378 

9.99 517 

28 

33 

9.17 223 

84 

9.17 708 

86 

10.82 292 

9.99 515 

27 

34 

9.17 307 

84 

Qi 

9.17 794 

86 

Cft 

10.82 206 

9.99 513 

26 

35 

9.17 391 


9.17 880 

oO 

10.82 120 

9.99 511 

25 

36 

9.17 474 

83 

9.17 965 

85 

10.82 035 

9.99 509 

24 

37 

9.17 558 

84 

9.18 051 

86 

10.81 949 

9.99 507 

23 

38 

9.17 641 

83 

9.18 136 

85 

10.81 864 

9.99 505 

22 

39 

9.17 724 

83 

9.18 221 

85 

CC 

10.81 779 

9.99 503 

21 

40 

9.17 807 

83 

9.18 306 

OO 

10.81 694 

9.99 501 

20 

41 

9.17 890 

83 

9.18 391 

85 

10.81 609 

9.99 499 

19 

42 

9.17 973 

83 

9.18 475 

84 

10.81 525 

9.99 497 

18 

43 

9.18 055 

82 

9.18 560 

85 

10.81 440 

9.99 495 

17 

44 

9.18 137 

82 

9.18 644 

84 

10.81 356 

9.99 494 

16 

45 

9.18 220 

83 

9.18 728 


10.81 272 

9.99 492 

15 

46 

9.18 302 

82 

9.18 812 

84 

10.81 188 

9.99 490 

14 

47 

9.18 383 

81 

9.18 896 

84 

10.81 104 

9.99 488 

13 

48 

9.18 465 

82 

9.18 979 

83 

10.81 021 

9.99 486 

12 

49 

9.18 547 

82 

9.19 063 

84 

OQ 

10.80 937 

9.99 484 

11 

50 

9.18 628 

81 

9.19 146 

OO 

10.80 854 

9.99 482 

10 

51 

9.18 709 

81 

9.19 229 

83 

10.80 771 

9.99 480 

9 

52 

9.18 790 

81 

9.19 312 

83 

10.80 688 

9.99 478 

8 

53 

9.18 871 

81 

9.19 395 

83 

10.80 605 

9.99 476 

7 

54 

9.18 952 

81 

9.19 478 

83 

OQ 

10.80 522 

9.99 474 

6 

55 

9.19 033 

81 

9.19 561 

OO 

10.80 439 

9.99 472 

5 

56 

9.19 113 

80 

9.19 643 

82 

10.80 357 

9.99 470 

4 

57 

9.19 193 

80 

9.19 725 

82 

10.80 275 

9.99 468 

3 

58 

9.19 273 

80 

9.19 807 

82 

10.80 193 

9.99 466 

2 

59 

9.19 353 

80 

9.19 889 

82 

oo 

10.80 111 

9.99 464 

1 

60 

9.19 433 

80 

9.19 971 

OA 

10.80 029 

9.99 462 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 


Prop. Pts. 


" 

92 

91 

6 

9.2 

9.1 

7 

10.7 

10.6 

8 

12.3 

12.1 

9 

13.8 

13.6 

10 

15.3 

15.2 

20 

30.7 

30.3 

30 

46.0 

45.5 

40 

61.3 

60.7 

50 

76.7 

75.8 

// 

89 

88 

6 

8.9 

8.8 

7 

10.4 

10.3 

8 

11.9 

11.7 

9 

13.4 

13.2 

10 

14.8 

14.7 

20 

29.7 

29.3 

30 

44.5 

44.0 

40 

59.3 

58.7 

50 

74.2 

73.3 

// 

86 

85 

6 

8.6 

8.5 

7 

10.0 

9.9 

8 

11.5 

11.3 

9 

12.9 

12.8 

10 

14.3 

14.2 

20 

28.7 

28.3 

30 

43.0 

42.5 

40 

57.3 

56.7 

50 

71.7 

70.8 


// 

83 

82 

6 

8.3 

8.2 

7 

9.7 

9.6 

8 

11.1 

10.9 

9 

12.4 

12.3 

10 

13.8 

13.7 

20 

27.7 

27.3 

30 

41.5 

41.0 

40 

55.3 

54.7 

50 

69.2 

68.3 


" 

80 

2 

6 

8.0 

0.2 

7 

9.3 

0.2 

8 

10.7 

0.3 

9 

12.0 

0.3 

10 

13.3 

0.3 

20 

26.7 

0.7 

30 

40.0 

1.0 

40 

53.3 

1.3 

50 

66.7 

1.7 


90 

9.0 

10.5 
12.0 

13.5 
15.0 
30.0 
45.0 
60.0 
75.0 


87 

8.7 

10.2 

11.6 

13.0 

14.5 
29.0 

43.5 
58.0 

72.5 


84 

8.4 

9.8 

11.2 

12.6 

14.0 

28.0 

42.0 

56.0 

70.0 


81 

8.1 

9.4 

10.8 

12.2 

13.5 
27.0 

40.5 
54.0 

67.5 


0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.5 

0.7 

0.8 


Prop. Pts. 


81 ' 



































































36 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


9° 


, 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

o 

9.19 433 


9.19 971 

82 

10.80 029 

9.99 462 

60 





1 

9.19 513 

80 

9.20 053 

10.79 947 

9.99 460 

59 





2 

9.19 592 

79 

9.20 134 

81 

10.79 866 

9.99 458 

58 





3 

9.19 672 

80 

9.20 216 

J82 

10.79 784 

9.99 456 

57 





4 

9.19 751 

79 

9.20 297 

81 

10.79 703 

9.99 454 

56 





5 

9.19 830 


9.20 378 


10.79 622 

9.99 452 

55 



79 

78 

6 

9.19 909 

79 

9.20 459 

81 

10.79 541 

9.99 450 

54 

" 

80 

7 

9.19 988 

79 

9.20 540 

81 

10.79 460 

9.99 448 

53 

a 

fi n 

n o 

7 8 

8 

9.20 067 

79 

9.20 621 

81 

10.79 379 

9.99 446 

52 

D 

7 

o.U 

9.3 

/ .y 
9.2 

9.1 

9 

9.20 145 

78 

9.20 701 

80 

10.79 299 

9.99 444 

51 

8 

10.7 

10.5 

10.4 

10 

9.20 223 


9.20 782 

80 

10.79 218 

9.99 442 

50 

9 

1 A 

12.0 

no 

11.8 
10 0 

11.7 

1 5 n 

11 

9.20 302 

79 

9.20 862 

10.79 138 

9.99 440 

49 

IU 

20 

lo.o 

26 7 

LO.Z 

26 3 

IO.U 

26.0 

12 

9.20 380 

78 

9.20 942 

80 

10.79 058 

9.99 438 

48 

30 

40.0 

39.5 

39.0 

13 

9.20 458 

78 

9.21 022 

80 

10.78 978 

9.99 436 

47 

40 

53.3 

52.7 

52.0 

14 

9.20 535 

77 

9.21 102 

80 

10.78 898 

9.99 434 

46 

50 

66.7 

65.8 

65.0 

15 

9.20 613 


9.21 182 

79 

10.78 818 

9.99 432 

45 





16 

9.20 691 

78 

9.21 261 

10.78 739 

9.99 429 

44 





17 

9.20 768 

77 

9.21 341 

80 

10.78 659 

9.99 427 

43 





18 

9.20 845 

77 

9.21 420 

79 

10.78 580 

9.99 425 

42 





19 

9.20 922 

77 

9.21 499 

79 

10.78 501 

9.99 423 

41 

„ 

77 

76 

75 

20 

9.20 999 


9.21 578 

79 

10.78 422 

9.99 421 

40 


7.6 


21 

9.21 076 

77 

9.21 657 

10.78 343 

9.99 419 

39 

6 

7.7 

7.5 

22 

9.21 153 

77 

9.21 736 

79 

10.78 264 

9.99 417 

38 

7 

Q 

9.0 

8.9 

ini 

8.8 

inn 

23 

9.21 229 

76 

9.21 814 

78 

10.78 186 

9.99 415 

37 

O 

9 

116 

1 U .1 

11.4 

1U.U 

11.2 

24 

9.21 306 

77 

9.21 893 

79 

10.78 107 

9.99 413 

36 

10 

12.8 

12.7 

12.5 

25 

26 

9.21 382 
9.21 458 

76 

76 

9.21 971 

9.22 049 

/ 0 

78 

10.78 029 
10.77 951 

9.99 411 
9.99 409 

35 

34 

20 

30 

40 

25.7 
38.5 
51 3 

25.3 
38.0 
50 7 

25.0 

37.5 

50.0 

27 

9.21 534 

76 

9.22 127 

78 

10.77 873 

9.99 407 

33 

50 

64.2 

63.3 

62.5 

28 

9.21 610 

76 

9.22 205 

78 

10.77 795 

9.99 404 

32 




29 

9.21 685 

75 

9.22 283 

78 

78 

10.77 717 

9.99 402 

31 





30 

9.21 761 

76 

9.22 361 

1 0 

77 

10.77 639 

9.99 400 

30 





31 

9.21 836 

75 

9.22 438 

10.77 562 

9.99 398 

29 





32 

9.21 912 

76 

9.22 516 

78 

10.77 484 

9.99 396 

28 


74 

73 

72 

33 

9.21 987 

75 

9.22 593 

77 

10.77 407 

9.99 394 

27 

" 

34 

9.22 062 

75 

7C 

9.22 670 

77 

77 

10.77 330 

9.99 392 

26 

6 

7.4 

7.3 

7.2 

35 

9.22 137 

/ o 

9.22 747 

77 

10.77 253 

9.99 390 

25 

7 

8.6 

8.5 

8.4 

36 

9.22 211 

74 

9.22 824 

10.77 176 

9.99 388 

24 

8 

9.9 

9.7 

9.6 

37 

9.22 286 

75 

9.22 901 

77 

10.77 099 

9.99 385 

23 

9 

1 n 

11.1 
12 3 

11.0 

12.2 

10.8 

12 0 

38 

9.22 361 

75 

9.22 977 

76 

10.77 023 

9.99 383 

22 

Ml 

20 

24^7 

24i3 

24!o 

39 

9.22 435 

74 

17 A 

9.23 054 

77 

7R 

10.76 946 

9.99 381 

21 

30 

37.0 

36.5 

36.0 

40 

9.22 509 

i *± 

9.23 130 

76 

10.76 870 

9.99 379 

20 

40 

CO 

49.3 

61.7 

48.7 
nn a 

48.0 
ah n 

41 

9.22 583 

74 

9.23 206 

10.76 794 

9.99 377 

19 

Otr 


OU.o 

uu.u 

42 

9.22 657 

74 

9.23 283 

77 

10.76 717 

9.99 375 

18 





43 

9.22 731 

74 

9.23 359 

76 

10.76 641 

9.99 372 

17 





44 

9.22 805 

74 

7Q 

9.23 435 

76 

7c; 

10.76 565 

9.99 370 

16 





45 

9.22 878 

/ O 

9.23 510 

76 

10.76 490 

9.99 368 

15 





46 

9.22 952 

74 

9.23 586 

10.76 414 

9.99 366 

14 

" 

71 

3 

2 

47 

9.23 025 

73 

9.23 661 

75 

10.76 339 

9.99 364 

13 

6 




48 

9.23 098 

73 

9.23 737 

76 

10.76 263 

9.99 362 

12 

7.1 

0.3 

0.2 

49 

9.23 171 

73 

7Q 

9.23 812 

75 

' 7 E 

10.76 188 

9.99 359 

11 

7 

8 

8.3 

9.5 

0.4 

0.4 

0.2 

0.3 

50 

9.23 244 

/ O 

9.23 887 

4 O 

75 

10.76 113 

9.99 357 

10 

9 

10.6 

0.4 

0.3 

51 

9.23 317 

73 

9.23 962 

10.76 038 

9.99 355 

9 

10 

11.8 

0.5 

0.3 

52 

9.23 390 

73 

9.24 037 

75 

10.75 963 

9.99 353 

8 

20 

on 

23.7 

QC C 

1.0 

1 k 

0.7 

1 0 

53 

9.23 462 

72 

9.24 112 

75 

10.75 888 

9.99 351 

7 

OU 

40 

oO.O 

47.3 

1.0 

2 0 

L3 

54 

9.23 535 

73 

72 

9.24 186 

74 

75 

10.75 814 

9.99 348 

6 

50 

59.2 

2.5 

1.7 

55 

9.23 607 

9.24 261 


10.75 739 

9.99 346 

5 





56 

9.23 679 

72 

9.24 335 

74 

10.75 665 

9 99 344 

4 





57 

9.23 752 

73 

9.24 410 

75 

10.75 590 

y.99 342 

3 





58 

9.23 823 

71 

9.24 484 

74 

10.75 516 

9.99 340 

2 





59 

9.23 895 

72 

72 

9.24 558 

74 

10.75 442 

9.99 337 

1 





60 

9.23 967 

9.24 632 

4 4 

10.75 368 

9.99 335 

0 





1 _ 

L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 


Prop. Pts 



80 ‘ 






































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


37 


10 ° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

0 

9.23 967 


9.24 632 


10.75 368 

9.99 335 

60 





1 

9.24 039 

72 

9.24 706 

74 

10.75 294 

9.99 333 

59 





2 

9.24 110 

71 

9.24 779 

73 

10.75 221 

9.99 331 

58 





3 

9.24 181 

71 

9.24 853 

74 

10.75 147 

9.99 328 

57 





4 

9.24 253 

72 

9.24 926 

73 

74 

10.75 074 

9.99 326 

56 





5 

9.24 324 


9.25 000 

10.75 000 

9.99 324 

55 

// 

74 

73 

72 

6 

9.24 395 

71 

9.25 073 

73 

10.74 927 

9.99 322 

54 

g 

7 4 

7 3 

7 2 

7 

9.24 466 

71 

9.25 146 

73 

10.74 854 

9.99 319 

53 

7 

8!6 

8^5 

8.4 

8 

9.24 536 

70 

9.25 219 

73 

10.74 781 

9.99 317 

52 

8 

9.9 

9.7 

9.6 

9 

9.24 607 

71 

9.25 292 

73 

10.74 708 

99.9 315 

51 

9 

11.1 

11.0 

10.8 









10 

20 

12.3 

24.7 

12.2 

24.3 

12.0 

24.0 

10 

9.24 677 


9.25 365 

ro 

10.74 635 

9.99 313 

50 

11 

9.24 748 

71 

9.25 437 

72 

10.74 563 

9.99 310 

49 

30 

37.0 

36.5 

36.0 

12 

9.24 818 

70 

9.25 510 

73 

10.74 490 

9.99 308 

48 

40 

49.3 

48.7 

48.0 

13 

9.24 888 

70 

9.25 582 

72 

10.74 418 

9.99 306 

47 

50 

61.7 

60.8 

60.0 

14 

9.24 958 

70 

9.25 655 

73 

10.74 345 

9.99 304 

46 





15 

9.25 028 


9.25 727 


10.74 273 

9.99 301 

45 





16 

9.25 098 

70 

9.25 799 

72 

10.74 201 

9.99 299 

44 





17 

9.25 168 

70 

9.25 871 

72 

10.74 129 

9.99 297 

43 





18 

8.25 237 

69 

9.25 943 

72 

10.74 057 

9.99 294 

42 





19 

9.25 307 

70 

9.26 015 

72 

10.73 985 

9.99 292 

41 

" 

71' 

70 

69 

20 

9.25 376 


9.26 086 


10.73 914 

9.99 290 

40 



7 n 

R Q 

21 

9.25 445 

69 

9.26 158 

72 

10.73 842 

9.99 288 

39 

o 

7 

/ .1 
8 3 

1 .u 
8.2 

o.y 

8.0 

22 

9.25 514 

69 

9.26 229 

71 

10.73 771 

9.99 285 

38 

8 

9.5 

9.3 

9.2 

23 

9.25 583 

69 

9.26 301 

72 

10.73 699 

9.99 283 

37 

9 

10.6 

10.5 

10.4 

24 

9.25 652 

69 

9.26 372 

71 

10.73 628 

9.99 281 

36 

10 

20 

11.8 
23 7 

11.7 
23 3 

11.5 

23 0 

25 

9.25 721 

oy 

9.26 443 


10.73 557 

9.99 278 

35 

30 

35.5 

35i0 

34.5 

26 

9.25 790 

69 

9.26 514 

71 

10.73 486 

9.99 276 

34 

40 

47.3 

46.7 

46.0 

27 

9.25 858 

68 

9.26 585 

71 

10.73 415 

9.99 274 

33 

50 

59.2 

58.3 

57.5 

28 

9.25 927 

69 

9.26 655 

70 

10.73 345 

9.99 271 

32 





29 

9.25 995 

68 

9.26 726 

71 

71 

10.73 274 

9.99 269 

31 





30 

9.26 063 

Do 

9.26 797 


10.73 203 

9.99 267 

30 





31 

9.26 131 

68 

9.26 867 

70 

10.73 133 

9.99 264 

29 





32 

9.26 199 

68 

9.26 937 

70 

10.73 063 

9.99 262 

28 





33 

9.26 267 

68 

9.27 008 

71 

10.72 992 

9.99 260 

27 

// 

68 

67 

66 

34 

9.26 335 

68 

9.27 078 

70 

7n 

10.72 922 

9.99 257 

26 


6.8 
7 9 

6.7 
7 8 


35 

9.26 403 

68 

9.27 148 


10.72' 852 

9.99 255 

25 

6 

7 

6.6 

7 7 

36 

9.26 470 

67 

9.27 218 

70 

10.72 782 

9.99 252 

24 

8 

9il 

8^9 

8.8 

37 

9.26 538 

68 

9.27 288 

70 

10.72 712 

9.99 250 

23 

9 

10.2 

10.0 

9.9 

38 

9.26 605 

67 

9.27 357 

69 

10.72 643 

9.99 248 

22 

10 

11.3 

11.2 

11.0 

39 

9.26 672 

67 

9.27 427 

70 

69 

10.72 573 

9.99 245 

21 

20 

30 

22.7 
34 0 

22.3 

33.5 

22.0 

33.0 

40 

9.26 739 

67 

9.27 496 


10.72 504 

9.99 243 

20 

40 

45.3 

44.7 

44.0 

41 

9.26 806 

67 

9.27 566 

70 

10.72 434 

9.99 241 

19 

50 

56.7 

55.8 

55.0 

42 

9.26 873 

67 

9.27 635 

69 

10.72 365 

9.99 238 

18 





43 

9.26 940 

67 

9.27 704 

69 

10.72 296 

9.99 236 

17 





44 

9.27 007 

67 

9.27 773 

69 

69 

10.72T 227 

9.99 233 

16 





45 

9.27 073 

66 

9.27 842 


10.72 158 

9.99 231 

15 





46 

9.27 140 

67 

9.27 911 

69 

10.72 089 

9.99 229 

14 





47 

9.27 206 

66 

9.27 980 

69 

10.72 020 

9.99 226 

13 

„ 

65 

3 

2 

48 

9.27 273 

67 

9.28 049 

69 

10.71 951 

9.99 224 

12 





49 

9.27 339 

66 

9.28 117 

68 

R Q 

10.71 883 

9.99 221 

11 

6 

6.5 

0.3 

n a 

0.2 
n 9 

50 

9.27 405 

66 

9.28 186 

oy 

10.71 814 

9.99 219 

10 

7 

8 

7.6 

8.7 

U .4 

0.4 

VJ . a , 

0.3 

51 

9.27 471 

66 

9.28 254 

68 

10.71 746 

9.99 217 

9 

9 

9.8 

0.4 

0.3 

52 

9.27 537 

66 

9.28 323 

69 

10.71 677 

9.99 214 

8 

10 

10.8 

0.5 

0.3 

53 

9 27 602 

65 

9.28 391 

68 

10.71 609 

9.99 212 

7 

20 

21.7 

1.0 

0.7 

54 

9!27 668 

66 

9.28 459 

68 

CO 

10.71 541 

9.99 209 

6 

30 

40 

32.5 

43.3 

1.5 

2.0 

1.0 

1.3 

55 

9.27 734 

66 

9.28 527 

Do 

10.71 473 

9.99 207 

5 

50 

54.2 

2.5 

1.7 

56 

9.27 799 

65 

9.28 595 

68 

10.71 405 

9.99 204 

4 





57 

9.27 864 

65 

9.28 662 

67 

10.71 338 

9.99 202 

3 





58 

9.27 930 

66 

9.28 730 

68 

10.71 270 

9.99 200 

2 





59 

9.27 995 

65 

9.28 798 

68 

10.71 202 

9.99 197 

1 





60 

9.28 060 

65 

9.28 865 

67 

10.71 135 

9.99 195 

0 






L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


79 ' 









































































38 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


11 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 



Prop. Pts. 


0 

9.28 060 

65 

9.28 865 

68 

10.71 135 

9.99 

195 

60 





1 

9.28 125 

9.28 933 

10.71 067 

9.99 

192 

59 





2 

9.28 190 

65 

9.29 000 

67 

10.71 000 

9.99 

190 

58 





3 

9.28 254 

64 

9.29 067 

67 

10.70 933 

9.99 

187 

57 





4 

9.28 319 

65 

9.29 134 

67 

10.70 866 

9.99 

185 

56 



67 

66 

5 

9.28 384 


9.29 201 

67 

10.70 799 

9.99 

182 

55 

" 

68 

6 

9.28 448 

64 

9.29 268 

10.70 732 

9.99 

180 

54 

6 

6.8 

6.7 

6.6 

7 

9.28 512 

64 

9.29 335 

67 

10.70 665 

9.99 

177 

53 

7 

7.9 

7.8 

7.7 

8 

9.28 577 

65 

9.29 402 

67 

10.70 598 

9.99 

175 

52 

8 

9.1 

8.9 

8.8 

9 

9.28 641 

64 

9.29 468 

66 

10.70 532 

9.99 

172 

51 

9 

10.2 
11 3 

10.0 
11 2 

9.9 

11 0 

10 

9.28 705 


9.29 535 

66 

10.70 465 

9.99 

170 

50 

20 

22.7 

22^3 

22^0 

11 

9.28 769 

64 

9.29 601 

10.70 399 

9.99 

167 

49 

30 

34.0 

33.5 

33.0 

12 

9.28 833 

64 

9.29 668 

67 

10.70 332 

9.99 

165 

48 

40 

45.3 

44.7 

44.0 

13 

9.28 896 

63 

9.29 734 

66 

10.70 266 

9.99 

162 

47 

50 

56.7 

55.8 

55.0 

14 

9.28 960 

64 

9.29 800 

66 

10.70 200 

9.99 

160 

46 





15 

9.29 024 


9.29 866 


10.70 134 

9.99 

157 

45 





16 

9.29 087 

63 

9.29 932 

66 

10.70 068 

9.99 

155 

44 





17 

9.29 150 

63 

9.29 998 

66 

10.70 002 

9.99 

152 

43 





18 

9.29 214 

64 

9.30 064 

66 

10.69 936 

9.99 

150 

42 





19 

9.29 277 

63 

9.30 130 

66 

10.69 870 

9.99 

147 

41 

" 

65 

64 

63 

20 

9.29 340 


9.30 195 


10.69 805 

9.99 

145 

40 

A 


A A 

A 9 

21 

9.29 403 

63 

9.30 261 

66 

10.69 739 

9.99 

142 

39 

D 

7 

6 5 
7^6 

7.5 

O.o 

7.4 

22 

9.29 466 

63 

9.30 326 

65 

10.69 674 

9.99 

140 

38 

8 

8.7 

8.5 

8.4 

23 

9.29 529 

63 

9.30 391 

65 

10.69 609 

9.99 

137 

37 

9 

9.8 

9.6 

9.4 

24 

9.29 591 

62 

AQ 

9.30 457 

66 

ac 

10.69 543 

9.99 

135 

36 

10 

10.8 
91 7 

10.7 
21 3 

10.5 

21 0 

25 

9.29 654 

Do 

9.30 522 

Ud 

65 

10.69 478 

9.99 

132 

35 

30 

32.5 

32i0 

3l!5 

26 

9.29 716 

62 

9.30 587 

10.69 413 

9.99 

130 

34 

40 

43.3 

42.7 

42.0 

27 

9.29 779 

63 

9.30 652 

65 

10.69 348 

9.99 

127 

33 

50 

54.2 

53.3 

52.5 

28 

9.29 841 

62 

9.30 717 

65 

10.69 283 

9.99 

124 

32 





29 

9.29 903 

62 

A9 

9.30 782 

65 

AJ. 

10.69 218 

9.99 

122 

31 





30 

9.29 966 

Oo 

9.30 846 

65 

10.69 154 

9.99 

119 

30 





31 

9.30 028 

62 

9.30 911 

10.69 089 

9.99 

117 

29 





32 

9.30 090 

62 

9.30 975 

64 

10.69 025 

9.99 

114 

28 





33 

9.30 151 

61 

9.31 040 

65 

10.68 960 

9.99 

112 

27 

„ 

62 

61 

60 

34 

9.30 213 

62 

no 

9.31 104 

64 

10.68 896 

9.99 

109 

26 



35 

9.30 275 

oz 

9.31 168 

65 

10.68*832 

9.99 

106 

25 

6 

n 

6.2 
7 9 

6.1 

6 ro 

36 

9.30 336 

61 

9.31 233 

10.68 767 

9.99 

104 

24 

/ 

8 

t .A 

8.3 

si 1 

8.o 

37 

9.30 398 

62 

9.31 297 

64 

10.68 703 

9.99 

101 

23 

9 

9.3 

9.2 

9.0 

38 

9.30 459 

61 

9.31 361 

64 

10.68 639 

9.99 

099 

22 

10 

10.3 

10.2 

10.0 

39 

9.30 521 

62 

A1 

9.31 425 

64 

p,A 

10.68 575 

9.99 

096 

21 

20 

9f> 

20.7 

9i n 

20.3 

9n 

20.0 

30 0 

40 

9.30 582 

01 

9.31 489 

63 

10.68 511 

9.99 

093 

20 

OU 

40 

Ol .u 

41.3 

OU.d 

40.7 

40^0 

41 

9.30 643 

61 

9.31 552 

10.68 448 

9.99 

091 

19 

50 

51.7 

50.8 

50.0 

42 

9.30 704 

61 

9.31 616 

64 

10.68 384 

9.99 

088 

18 





43 

9.30 765 

61 

9.31 679 

63 

10.68 321 

9.99 

086 

17 





44 

9.30 826 

61 

A1 

9.31 743 

64 

A9 

10.68 257 

9.99 

083 

16 





45 

9.30 887 

01 

9.31 806 

Do 

64 

10.68 194 

9.99 

080 

15 





46 

9.30 947 

60 

9.31 870 

10.68 130 

9.99 

078 

14 





47 

9.31 008 

61 

9.31 933 

63 

10.68 067 

9.99 

075 

13 

n 

RQ 

9 

2 

48 

9.31 068 

60 

9.31 996 

63 

10.68 004 

9.99 

072 

12 


UJ 

o 


49 

9.31 129 

61 

60 

9.32 059 

63 

A9 

10.67 941 

9.99 

070 

11 

6 

5.9 

0.3 

0.2 

50 

9.31 189 

9.32 122 

Do 

63 

10.67 878 

9.99 

067 

10 

7 

6.9 

0.4 
n A 

0.2 

51 

9.31 250 

61 

9.32 185 

10.67 815 

9.99 

064 

9 

9 

8 8 

0 5 

o!3 

52 

9.31 310 

60 

9.32 248 

63 

10.67 752 

9.99 

062 

8 

10 

9.8 

0.5 

0.3 

53 

9.31 370 

60 

9.32 311 

63 

10.67 689 

9.99 

059 

7 

20 

19.7 

1.0 

0.7 

54 

9.31 430 

60 

60 

9.32 373 

62 

10.67 627 

9.99 

056 

6 

30 

29.5 

9Q 9 

1.5 

9 n 

1.0 
i 9 

55 

9.31 490 

9.32 436 

62 

10.67 564 

9.99 

054 

5 

50 

oy .o 
49.2 

2.5 

1 .o 

1.7 

56 

9.31 549 

59 

9.32 498 

10.67 502 

9.99 

051 

4 



57 

9.31 609 

60 

9.32 561 

63 

10.67 439 

9.99 

048 

3 





58 

9.31 669 

60 

9.32 623 

62 

10.67 377 

9.99 

046 

2 





59 

9.31 728 

59 

An 

9.32 685 

62 

10.67 315 

9.99 

043 

1 





60 

9.31 788 

DU 

9.32 747 

62 

10.67 253 

9.99 

040 

0 






L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 

Prop. Pts. 


78 ‘ 









































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


39 


12 c 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


0 

9.31 788 


9.32 747 


10.67 253 

9.99 040 

60 

1 

9.31 847 

59 

9.32 810 

63 

10.67 190 

9.99 038 

59 

2 

9.31 907 

60 

9.32 872 

62 

10.67 128 

9.99 035 

58 

3 

9.31 966 

59 

9.32 933 

61 

10.67 067 

9.99 032 

57 

4 

9.32 025 

59 

59 

9.32 995 

62 

62 

10.67 005 

9.99 030 

56 

5 

9.32 084 

9.33 057 

10.66 943 

9.99 027 

55 

6 

9.32 143 

59 

9.33 119 

62 

10.66 881 

9.99 024 

54 

7 

9.32 202 

59 

9.33 180 

61 

10.66 820 

9.99 022 

53 

8 

9.32 261 

59 

9.33 242 

62 

10.66 758 

9.99 019 

52 

9 

9.32 319 

58 

59 

9.33 303 

61 

62 

10.66 697 

9.99 016 

51 

10 

9.32 378 

9.33 365 

10.66 635 

9.99 013 

50 

11 

9.32 437 

59 

9.33 426 

61 

10.66 574 

9.99 011 

49 

12 

9.32 495 

58 

9.33 487 

61 

10.66 513 

9.99 008 

48 

13 

9.32 553 

58 

9.33 548 

61 

10.66 452 

9.99 005 

47 

14 

9.32 612 

59 

58 

9.33 609 

61 

61 

10.66 391 

9.99 002 

46 

15 

9.32 670 

9.33 670 

10.66 330 

9.99 000 

45 

16 

9.32 728 

58 

9.33 731 

61 

10.66 269 

9.98 997 

44 

17 

9.32 786 

58 

9.33 792 

61 

10.66 208 

9.98 994 

43 

18 

9.32 844 

58 

9.33 853 

61 

10.66 147 

9.98 991 

42 

19 

9.32 902 

58 

58 

9.33 913 

60 

61 

10.66 087 

9.98 989 

41 

20 

9.32 960 

9.33 974 

10.66 026 

9.98 986 

40 

21 

9.33 018 

58 

9.34 034 

60 

10.65 966 

9.98 983 

39 

22 

9.33 075 

57 

9.34 095 

61 

10.65 905 

9.98 980 

38 

23 

9.33 133 

58 

9.34 155 

60 

10.65 845 

9.98 978 

37 

24 

9.33 190 

57 

58 

9.34 215 

60 

61 

10.65 785 

9.98 975 

36 

25 

9.33 248 

9.34 276 

10.65 724 

9.98 972 

35 

26 

9.33 305 

57 

9.34 336 

60 

10.65 664 

9.98 969 

34 

27 

9.33 362 

57 

9.34 396 

60 

10.65 604 

9.98 967 

33 

28 

9.33 420 

58 

9.34 456 

60 

10.65 544 

9.98 964 

32 

29 

9.33 477 

57 

57 

9.34 516 

60 

60 

10.65 484 

9.98 961 

31 

30 

9.33 534 

9.34 576 

10.65 424 

9.98 958 

30 

31 

9.33 591 

57 

9.34 635 

59 

10.65 365 

9.98 955 

29 

32 

9.33 647 

56 

9.34 695 

60 

10.65 305 

9.98 953 

28 

33 

9.33 704 

57 

9.34 755 

60 

10.65 245 

9.98 950 

27 

34 

9.33 761 

57 

9.34 814 

59 

60 

10.65 186 

9.98 947 

26 

35 

9.33 818 

O I 

9.34 874 

10.65 126 

9.98 944 

25 

36 

9.33 874 

56 

9.34 933 

59 

10.65 067 

9.98 941 

24 

37 

9.33 931 

57 

9.34 992 

59 

10.65 008 

9.98 938 

23 

38 

9.33 987 

56 

9.35 051 

59 

10.64 949 

9.98 936 

22 

39 

9.34 043 

56 

57 

9.35 111 

60 

10.64 889 

9.98 933 

21 

40 

9.34 100 

9.35 170 

59 

10.64 830 

9.98 930 

20 

41 

9.34 156 

56 

9.35 229 

59 

10.64 771 

9.98 927 

19 

42 

9.34 212 

56 

9.35 288 

59 

10.64 712 

9.98 924 

18 

43 

9.34 268 

56 

9.35 347 

59 

10.64 653 

9.98 921 

17 

44 

9.34 324 

56 

cc 

9.35 405 

58 

EQ 

10.64 595 

9.98 919 

16 

45 

9.34 380 

OD 

9.35 464 

oy 

10.64 536 

9.98 916 

15 

46 

9.34 436 

56 

9.35 523 

59 

10.64 477 

9.98 913 

14 

47 

9.34 491 

55 

9.35 581 

58 

10.64 419 

9.98 910 

13 

48 

9.34 547 

56 

9.35 640 

59 

10.64 360 

9.98 907 

12 

49 

9.34 602 

55 

9.35 698 

58 

EQ 

10.64 302 

9.98 904 

11 

50 

9.34 658 

56 

9.35 757 

oy 

10.64 243 

9.98 901 

10 

51 

9.34 713 

55 

9.35 815 

58 

10.64 185 

9.98 898 

9 

52 

9.34 769 

56 

9.35 873 

58 

10.64 127 

9.98 896 

8 

53 

9.34 824 

55 

9.35 931 

58 

10.64 069 

9.98 893 

7 

54 

9.34 879 

55 

9.35 989 

58 

co 

10.64 011 

9.98 890 

6 

55 

9.34 934 

55 

9.36 047 

oo 

10.63 953 

9.98 887 

5 

56 

9.34 989 

55 

9.36 105 

58 

10.63 895 

9.98 884 

4 

57 

9.35 044 

55 

9.36 163 

58 

10.63 837 

9.98 881 

3 

58 

9.35 099 

55 

9.36 221 

58 

10.63 779 

9.98 878 

2 

59 

9.35 154 

55 

9.36 279 

58 

10.63 721 

9.98 875 

1 

60 

9.35 209 

55 

9.36 336 

0 t 

10.63 664 

9.98 872 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 


Prop. Pts. 


" 

63 

62 

61 

6 

6.3 

6.2 

6.1 

7 

7.4 

7.2 

7.1 

8 

8.4 

8.3 

8.1 

9 

9.4 

9.3 

9.2 

10 

10.5 

10.3 

10.2 

20 

21.0 

20.7 

20.3 

30 

31.5 

31.0 

30.5 

40 

42.0 

41.3 

40.7 

50 

52.5 

51.7 

50.8 


" 

60 

59 

58 

6 

6.0 

5.9 

5.8 

7 

7.0 

6.9 

6.8 

8 

8.0 

7.9 

7.7 

9 

9.0 

8.8 

8.7 

10 

10.0 

9.8 

9.7 

20 

20.0 

19.7 

19.3 

30 

30.0 

29.5 

29.0 

40 

40.0 

39.3 

38.7 

50 

50.0 

49.2 

48.3 


" 

57 

56 

55 

6 

5.7 

5.6 

5.5 

7 

6.6 

6.5 

6.4 

8 

7.6 

7.5 

7.3 

9 

8.6 

8.4 

8.2 

10 

9.5 

9.3 

9.2 

20 

19.0 

18.7 

18.3 

30 

28.5 

28.0 

27.5 

40 

38.0 

37.3 

36.7 

50 

47.5 

46.7 

45.8 


" 

3 

2 

6 

0.3 

0.2 

7 

0.4 

0.2 

8 

0.4 

0.3 

9 

0.5 

0.3 

10 

0.5 

0.3 

20 

1.0 

0.7 

30 

1.5 

1.0 

40 

2.0 

1.3 

50 

2.5 

1.7 


Proo. Pts. 


77 ' 




































































40 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


13 c 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


0 

9.35 209 

54 

9.36 336 

58 

10.63 

664 

9.98 

872 

60 

1 

9.35 263 

9.36 394 

10.63 

606 

9.98 

869 

59 

2 

9.35 318 

55 

9.36 452 

58 

10.63 

548 

9.98 

867 

58 

3 

9.35 373 

55 

9.36 509 

57 

10.63 

491 

9.98 

864 

57 

4 

9.35 427 

54 

9.36 566 

57 

eq 

10.63 

434 

9.98 

861 

56 

5 

9.35 481 


9.36 624 

Oo 

57 

10.63 

376 

9.98 

858 

55 

6 

9.35 536 

55 

9.36 681 

10.63 

319 

9.98 

855 

54 

7 

9.35 590 

54 

9.36 738 

57 

10.63 

262 

9.98 

852 

53 

8 

9.35 644 

54 

9.36 795 

57 

10.63 

205 

9.98 

849 

52 

9 

9.35 698 

54 

9.36 852 

57 

E7 

10.63 

148 

9.98 

846 

51 

10 

9.35 752 


9.36 909 

O l 

57 

10.63 

091 

9.98 

843 

50 

11 

9.35 806 

54 

9.36 966 

10.63 

034 

9.98 

840 

49 

12 

9.35 860 

54 

9.37 023 

57 

10.62 

977 

9.98 

837 

48 

13 

9.35 914 

54 

9.37 080 

57 

10.62 

920 

9.98 

834 

47 

14 

9.35 968 

54 

9.37 137 

57 

56 

10.62 

863 

9.98 

831 

46 

15 

9.36 022 


9.37 193 


10.62 

807 

9.98 

828 

45 

16 

9.36 075 

53 

9.37 250 

57 

10.62 

750 

9.98 

825 

44 

17 

9.36 129 

54 

9.37 306 

56 

10.62 

694 

9.98 

822 

43 

18 

9.36 182 

53 

9.37 363 

57 

10.62 

637 

9.98 

819 

42 

19 

9.36 236 

54 

9.37 419 

56 

E7 

10.62 

581 

9.98 

816 

41 

20 

9.36 289 


9.37 476 

O I 

10.62 

524 

9.98 

813 

40 

21 

9.36 342 

53 

9.37 532 

56 

10.62 

468 

9.98 

810 

39 

22 

9.36 395 

53 

9.37 588 

56 

10.62 

412 

9.98 

807 

38 

23 

9.36 449 

54 

9.37 644 

56 

10.62 

356 

9.98 

804 

37 

24 

9.36 502 

53 

9.37 700 

56 

56 

10.62 

300 

9.98 

801 

36 

25 

9.36 555 

53 

9.37 756 


10.62 

244 

9.98 

798 

35 

26 

9.36 608 

53 

9.37 812 

56 

10.62 

188 

9.98 

795 

34 

27 

9.36 660 

52 

9.37 868 

56 

10.62 

132 

9.98 

792 

33 

28 

9.36 713 

53 

9.37 924 

56 

10.62 

076 

9.98 

789 

32 

29 

9.36 766 

53 

CO 

9.37 980 

56 

ee 

10.62 

020 

9.98 

786 

31 

30 

9.36 819 

Oo 

9.38 035 

oo 

56 

10.61 

965 

9.98 

783 

30 

31 

9.36 871 

52 

9.38 091 

10.61 

909 

9.98 

780 

29 

32 

9.36 924 

53 

9.38 147 

56 

10.61 

853 

9.98 

777 

28 

33 

9.36 976 

52 

9.38 202 

55 

10.61 

798 

9.98 

774 

27 

34 

9.37 028 

52 

CQ 

9.38 257 

55 

e« 

10.61 

743 

9.98 

771 

26 

35 

9.37 081 

Oo 

9.38 313 

oo 

55 

10.61 

687 

9.98 

768 

25 

36 

9.37 133 

52 

9.38 368 

10.61 

632 

9.98 

765 

24 

37 

9.37 185 

52 

9.38423 

55 

10.61 

577 

9.98 

762 

23 

38 

9.37 237 

52 

9.38 479 

56 

10.61 

521 

9.98 

759 

22 

39 

9.37 289 

52 

PO 

9.38 534 

55 

ee 

10.61 

466 

9.98 

756 

21 

40 

9.37 341 

O Zt 

9.38 589 

OO 

10.61 

411 

9.98 

753 

20 

41 

9.37 393 

52 

9.38 644 

55 

10.61 

356 

9.98 

750 

19 

42 

9.37 445 

52 

9.38 699 

55 

10.61 

301 

9.98 

746 

18 

43 

9.37 497 

52 

9.38 754 

55 

10.61 

246 

9.98 

743 

17 

44 

9.37 549 

52 

PI 

9.38 808 

54 

E E 

10.61 

192 

9.98 

740 

16 

45 

9.37 600 

ol 

9.38 863 

OO 

55 

10.61 

137 

9.98 

737 

15 

46 

9.37 652 

52 

9.38 918 

10.61 

082 

9.98 

734 

14 

47 

9.37 703 

51 

9.38 972 

54 

10.61 

028 

9.98 

731 

13 

48 

9.37 755 

52 

9.39 027 

55 

10.60 

973 

9.98 

728 

12 

49 

9.37 806 

51 

9.39 082 

55 

10.60 

918 

9.98 

725 

11 

50 

9.37 858 

52 

9.39 136 

Ot: 

54 

10.60 

864 

9.98 

722 

10 

51 

9.37 909 

51 

9.39 190 

10.60 

810 

9.98 

719 

9 

52 

9.37 960 

51 

9.39 245 

55 

10.60 

755 

9.98 

715 

8 

53 

9.38 011 

51 

9.39 299 

54 

10.60 

701 

9.98 

712 

7 

54 

9.38 062 

51 

C 1 

9.39 353 

54 

10.60 

647 

9.98 

709 

6 

55 

9.38 113 

Ol 

9.39 407 

Ot± 

54 

10.60 

593 

9.98 

706 

5 

56 

9.38 164 

51 

9.39 461 

10.60 

539 

9.98 

703 

4 

57 

9.38 215 

51 

9.39 515 

54 

10.60 

485 

9.98 

700 

3 

58 

9.38 266 

51 

9.39 569 

54 

10.60 

431 

9.98 

697 

2 

59 

9.38 317 

51 

9.39 623 

54 

10.60 

377 

9.98 

694 

1 

60 

9.38 368 

Ol 

9.39 677 

Otfc 

10.60 

323 

9.98 

690 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

' 


Prop. Pts. 


" 

58 

57 

56 

6 

5.8 

5.7 

5.6 

7 

6.8 

6.6 

6.5 

8 

7.7 

7.6 

7.5 

9 

8.7 

8.6 

8.4 

10 

9.7 

9.5 

9.3 

20 

19.3 

19.0 

18.7 

30 

29.0 

28.5 

28.0 

40 

38.7 

38.0 

37.3 

50 

48.3 

47.5 

46.7 


" 

55 

54 

53 

6 

5.5 

5.4 

5.3 

7 

6.4 

6.3 

6.2 

8 

7.3 

7.2 

7.1 

9 

8.2 

8.1 

8.0 

10 

9.2 

9.0 

8.8 

20 

18.3 

18.0 

17.7 

30 

27.5 

27.0 

26.5 

40 

36.7 

36.0 

35.3 

50 

45.8 

45.0 

44.2 


" 

52 

51 

4 

6 

5.2 

5.1 

0.4 

7 

6.1 

6.0 

0.5 

8 

6.9 

6.8 

0.5 

9 

7.8 

7.6 

0.6 

10 

8.7 

8.5 

0.7 

20 

17.3 

17.0 

1.3 

30 

26.0 

25.5 

2.0 

40 

34.7 

34.0 

2.7 

50 

43.3 

42.5 

3.3 


" 

3 

2 

6 

0.3 

0.2 

7 

0.4 

0.2 

8 

0.4 

0.3 

9 

0.4 

0.3 

10 

0.5 

0.3 

20 

1.0 

0.7 

30 

1.5 

1.0 

40 

2.0 

1.3 

50 

2.5 

1.7 


Prop. Pts. 


76 ' 







































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


41 


14° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 


Prop. Pts. 

0 

9.38 368 


9.39 677 


10.60 323 

9.98 690 

60 







1 

9.38 418 

50 

9.39 731 

54 

10.60 269 

9.98 687 

59 







2 

9.38 469 

51 

9.39 785 

54 

10.60 215 

9.98 684 

58 







3 

9.38 519 

50 

9.39 838 

53 

10.60 162 

9.98 681 

57 







4 

9.38 570 

51 

50 

9.39 892 

54 

53 

10.60 108 

9.98 678 

56 







5 

9.38 620 

9.39 945 

10.60 055 

9.98 675 

55 

" 

54 

53 

52 

6 

9.38 670 

50 

9.39 999 

54 

10.60 001 

9.98 671 

54 







7 

9.38 721 

51 

9.40 052 

53 

10.59 948 

9.98 668 

53 

6 

7 

o.* 

ft Q 


5.3 
6 2 

5.2 

6 1 

8 

9.38 771 

50 

9.40 106 

54 

10.59 894 

9.98 665 

52 

8 


7^2 


7.1 

6^9 

9 

9.38 821 

50 

50 

9.40 159 

53 

53 

10.59 841 

9.98 662 

51 

9 

8.1 


8.0 

7.8 

10 

9.38 871 

9.40 212 

10.59 788 

9.98 659 

50 

10 

20 

9.0 
18 0 

8.8 
17 7 

8.7 

17 3 

11 

9.38 921 

50 

9.40 266 

54 

10.59 734 

9.98 656 

49 

30 

27!o 

26^5 

26^0 

12 

9.38 971 

50 

9.40 319 

53 

10.59 681 

9.98 652 

48 

40 

36.0 

35.3 

34.7 

13 

9.39 021 

50 

9.40 372 

53 

10.59 628 

9.98 649 

47 

50 

45.0 

44.2 

43.3 

14 

9.39 071 

50 

9.40 425 

53 

53 

10.59 575 

9.98 646 

46 







15 

9.39 121 

ou 

9.40 478 

10.59 522 

9.98 643 

45 







16 

9.39 170 

49 

9.40 531 

53 

10.59 469 

9.98 640 

44 







17 

9.39 220 

50 

9.40 584 

53 

10.59 416 

9.98 636 

43 







18 

9.39 270 

50 

9.40 636 

52 

10.59 364 

9.98 633 

42 







19 

9.39 319 

49 

9.40 689 

53 

10.59 311 

9.98 630 

41 

" 

51 

50 

49 

20 

9.39 369 

OU 

9.40 742 


10.59 258 

9.98 627 

40 







21 

9.39 418 

49 

9.40 795 

53 

10.59 205 

9.98 623 

39 

6 

7 

O.i 
a n 

o.u 

c 0 

4.9 

22 

9.39 467 

49 

9.40 847 

52 

10.59 153 

9.98 620 

38 

8 

o.u 

6.8 

0.8 

6.7 

6.5 

23 

9.39 517 

50 

9.40 900 

53 

10.59 100 

9.98 617 

37 

9 

7.6 

7.5 

7.4 

24 

9.39 566 

49 

9.40 952 

52 

10.59 048 

9.98 614 

36 

10 

8.5 

8.3 

8.2 





53 




20 

30 





16.3 

24.5 

25 

9.39 615 

4y 

9.41 005 

10.58 995 

9.98 610 

35 

l ! .U 

25.5 

-LU. ! 

25.0 

26 

9.39 664 

49 

9.41 057 

52 

10.58 943 

9.98 607 

34 

40 

34.0 

33.3 

32.7 

27 

9.39 713 

49 

9.41 109 

52 

10.58 891 

9.98 604 

33 

50 

42.5 

41.7 

40.8 

28 

9.39 762 

49 

9.41 161 

52 

10.58 839 

9.98 601 

32 







29 

9.39 811 

49 

49 

9.41 214 

53 

10.58 786 

9.98 597 

31 







30 

9.39 860 

9.41 266 

04 

10.58 734 

9.98 594 

30 







31 

9.39 909 

49 

9.41 318 

52 

10.58 682 

9.98 591 

29 







32 

9.39 958 

49 

9.41 370 

52 

10.58 630 

9.98 588 

28 







33 

9.40 006 

48 

9.41 422 

52 

10.58 578 

9.98 584 

27 


„ 

4ft 

47 

34 

9.40 055 

49 

A Q 

9.41 474 

52 

10.58 526 

9.98 581 

26 







35 

9.40 103 

4:0 

9.41 526 

04 

10.58 474 

9.98 578 

25 


6 

7 

4.8 

4.7 

C CL 

36 

9.40 152 

49 

9.41 578 

52 

10.58 422 

9.98 574 

24 


/ 

8 

0.0 
6 4 

0.0 

6 3 

37 

9.40 200 

48 

9.41 629 

51 

10.58 371 

9.98 571 

23 


9 

7.2 

7.0 

38 

9.40 249 

49 

9.41 681 

52 

10.58 319 

9.98 568 

22 

10 

8.0 

7.8 

39 

9.40 297 

48 

9.41 733 

52 

C 1 

10.58 267 

9.98 565 

21 

20 

qa 

16.0 
o a n 

15.7 

23 5 

40 

9.40 346 

49 

9.41 784 

01 

10.58 216 

9.98 561 

20 

ou 

40 

32.0 

3L3 

41 

9.40 394 

48 

9.41 836 

52 

10.58 164 

9.98 558 

19 

50 

40.0 

39.2 

42 

9.40 442 

48 

9.41 887 

51 

10.58 113 

9.98 555 

18 







43 

9.40 490 

48 

9.41 939 

52 

10.58 061 

9.98 551 

17 







44 

9.40 538 

48 

A Q 

9.41 990 

51 

ki 

10.58 010 

9.98 548 

16 







45 

9.40 586 


9.42 041 

O I 

10.57 959 

9.98 545 

15 







46 

9.40 634 

48 

9.42 093 

52 

10.57 907 

9.98 541 

14 







47 

9.40 682 

48 

9.42 144 

51 

10.57 856 

9.98 538 

13 


,, 

4 

[ 


[ 

48 

9.40 730 

48 

9.42 195 

51 

10.57 805 

9.98 535 

12 







49 

9.40 778 

48 

9.42 246 

51 

C 1 

10.57 754 

9.98 531 

11 


6 

0.4 

0.3 

50 

9.40 825 

47 

9.42 297 

Dl 

10.57 703 

9.98 528 

10 


7 

g 

0.5 

0 5 

0.4 

0 4 

51 

9.40 873 

48 

9.42 348 

51 

10.57 652 

9.98 525 

9 


9 

0^6 

o!4 

52 

9.40 921 

48 

9.42 399 

51 

10.57 601 

9.98 521 

8 

10 

0.7 

0.5 

53 

9.40 968 

47 

9.42 450 

51 

10.57 550 

9.98 518 

7 

20 

1.3 

1.0 

54 

9.41 016 

48 

9.42 501 

51 

51 

10.57 499 

9.98 515 

6 

30 

40 

2.0 

2 7 

1.5 

2.0 

55 

9.41 063 

47 

9.42 552 


10.57 448 

9.98 511 

5 

50 

3.3 

2.5 

56 

9.41 111 

48 

9.42 603 

51 

10.57 397 

9.98 508 

4 







57 

9.41 158 

47 

9.42 653 

50 

10.57 347 

9.98 505 

3 







58 

9.41 205 

47 

9.42 704 

51 

10.57 296 

9.98 501 

2 







59 

9.41 252 

47 

a e 

9.42 755 

51 

10.57 245 

9.98 498 

1 







60 

9.41 300 

45 

9.42 805 


10.57 195 

9.98 494 

0 








L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

r 

Prop. Pts. 


75 ( 

































































42 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


15 c 


> 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.41 300 


9.42 805 


10.57 195 

9.98 494 


60 


9.41 347 

47 

9.42 856 

51 

10.57 144 

9.98 491 

3 

59 

2 

9.41 394 

47 

9.42 906 

50 

10.57 094 

9.98 488 

3 

58 

3 

9.41 441 

47 

9.42 957 

51 

10.57 043 

9.98 484 

4 

57 

4 

9.41 488 

47 

9.43 007 

50 

cn 

10.56 993 

9.98 481 

3 

4 

56 

5 

9.41 535 


9.43 057 

ou 

10.56 943 

9.98 477 


55 

6 

9.41 582 

47 

9.43 108' 

51 

10.56 892 

9.98 474 

3 

54 

7 

9.41 628 

46 

9.43 158 - 

50 

10.56 842 

9.98 471 

3 

53 

8 

9.41 675 

47 

9.43 208 

50 

10.56 792 

9.98 467 

4 

52 

9 

9.41 722 

47 

9.43 258 

50 

10.56 742 

9.98 464 

3 

4 

51 

10 

9.41 768 


9.43 308 

OU 

10.56 692 

9.98 460 


50 

11 

9.41 815 

47 

9.43 358 

50 

10.56 642 

9.98 457 

3 

49 

12 

9.41 861 

46 

9.43 408 

50 

10.56 592 

9.98 453 

4 

48 

13 

9.41 908 

47 

9.43 458 

50 

10.56 542 

9.98 450 

3 

47 

14 

9.41 954 

46 

9.43 508 

50 

cn 

10.56 492 

9.98 447 

3 

4 

46 

15 

9.42 001 


9.43 558 

ou 

10.56 442 

9.98 443 


45 

16 

9.42 047 

46 

9.43 607 

49 

10.56 393 

9.98 440 

3 

44 

17 

9.42 093 

46 

9.43 657 

50 

10.56 343 

9.98 436 

4 

43 

18 

9.42 140 

47 

9.43 707 

50 

10.56 293 

9.98 433 

3 

42 

19 

9.42 186 

46 

9.43 756 

49 

10.56 244 

9.98 429 

4 

3 

41 

20 

9.42 232 


9.43 806 

ou 

10.56 194 

9.98 426 


40 

21 

9.42 278 

46 

9.43 855 

49 

10.56 145 

9.98 422 

4 

39 

22 

9.42 324 

46 

9.43 905 

50 

10.56 095 

9.98 419 

3 

38 

23 

9.42 370 

46 

9.43 954 

49 

10.56 046 

9.98 415 

4 

37 

24 

9.42 416 

46 

AK 

9.44 004 

50 

AQ 

10.55 996 

9.98 412 

3 

3 

36 

25 

9.42 461 


9.44 053 


10.55 947 

9.98 409 


35 

26 

9.42 507 

46 

9.44 102 

49 

10.55 898 

9.98 405 

4 

34 

27 

9.42 553 

46 

9.44 151 

49 

10.55 849 

9.98 402 

3 

33 

28 

9.42 599 

46 

9.44 201 

50 

10.55 799 

9.98 398 

4 

32 

29 

9.42 644 

45 

9.44 250 

49 

49 

10.55 750 

9.98 395 

3 

4 

31 

30 

9.42 690 

40 

9.44 299 

10.55 701 

9.98 391 


30 

31 

9.42 735 

45 

9.44 348 

49 

10.55 652 

9.98 388 

3 

29 

32 

9.42 781 

46 

9.44 397 

49 

10.55 603 

9.98 384 

4 

28 

33 

9.42 826 

45 

9.44 446 

49 

10.55 554 

9.98 381 

3 

27 

34 

9.42 872 

46 

9.44 495 

49 

49 

10.55 505 

9.98 377 

4 

4 

26 

35 

9.42 917 

40 

9.44 544 


10.55 456 

9.98 373 


25 

36 

9.42 962 

45 

9.44 592 

48 

10.55 408 

9.98 370 

3 

24 

37 

9.43 008 

46 

9.44 641 

49 

10.55 359 

9.98 366 

4 

23 

38 

9.43 053 

45 

9.44 690 

49 

10.55 310 

9.98 363 

3 

22 

39 

9.43 098 

45 

ax 

9.44 738 

48 

4Q 

10.55 262 

9.98 359 

4 

3 

21 

40 

9.43 143 

40 

9.44 787 


10.55 213 

9.98 356 


20 

41 

9.43 188 

45 

9.44 836 

49 

10.55 164 

9.98 352 

4 

19 

42 

9.43 233 

45 

9.44 884 

48 

10.55 116 

9.98 349 

3 

18 

43 

9.43 278 

45 

9.44 933 

49 

10.55 067 

9.98 345 

4 

17 

44 

9.43 323 

45 

AA 

9.44 981 

48 

as 

10.55 019 

9.98 342 

3 

A 

16 

45 

9.43 367 


9.45 029 


10.54 971 

9.98 338 

rt 

15 

46 

9.43 412 

45 

9.45 078 

49 

10.54 922 

9.98 334 

4 

14 

47 

9.43 457 

45 

9.45 126 

48 

10.54 874 

9.98 331 

3 

13 

48 

9.43 502 

45 

9.45 174 

48 

10.54 826 

9.98 327 

4 

12 

49 

9.43 546 

44 

45 

9.45 222 

48 

49 

10.54 778 

9.98 324 

3 

A 

11 

50 

9.43 591 

9.45 271 

10.54 729 

9.98 320 

t 

10 

51 

9.43 635 

44 

9.45 319 

48 

10.54 681 

9.98 317 

3 

9 

52 

9.43 680 

45 

9.45 367 

48 

10.54 633 

9.98 313 

4 

8 

53 

9.43 724 

44 

9.45 415 

48 

10.54 585 

9.98 309 

4 

7 

54 

9.43 769 

45 

44 

9.45 463 

48 

48 

10.54 537 

9.98 306 

3 

4 

6 

55 

9.43 813 

9.45 511 

10.54 489 

9.98 302 


5 

56 

9.43 857 

44 

9.45 559 

48 

10.54 441 

9.98 299 

3 

4 

57 

9.43 901 

44 

9.45 606 

47 

10.54 394 

9.98 295 

4 

3 

58 

9.43 946 

45 

9.45 654 

48 

10.54 346 

9.98 291 

4 

2 

59 

9.43 990 

44 

AA 

9.45 702 

48 

48 

10.54 298 

9.98 288 

3 

A 

1 

60 

9.44 034 


9.45 750 

10.54 250 

9.98 284 

i 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

/ 


Prop. Pts. 


" 

51 

50 

49 

6 

5.1 

5.0 

4.9 

7 

6.0 

5.8 

5.7 

8 

6.8 

6.7 

6.5 

9 

7.7 

7.5 

7.4 

10 

8.5 

8.3 

8.2 

20 

17.0 

16.7 

16.3 

30 

25.5 

25.0 

24.5 

40 

34.0 

33.3 

32.7 

50 

42.5 

41.7 

40.8 


" 

48 

47 

46 

6 

4.8 

4.7 

4.6 

7 

5.6 

5.5 

5.4 

8 

6.4 

6.3 

6.1 

9 

7.2 

7.0 

6.9 

10 

8.0 

7.8 

7.7 

20 

16.0 

15.7 

15.3 

30 

24.0 

23.5 

23.0 

40 

32.0 

31.3 

30.7 

50 

40.0 

39.2 

38.3 


45 

4.5 
5.3 
6.0 
6.8 

7.5 
15.0 

22.5 
30.0 

37.5 


44 

4.4 

5.1 

5.9 

6.6 

7.3 

14.7 
22.0 
29.3 

36.7 


0.3 

0.4 

0.4 

0.5 

0.5 

1.0 

1.5 
2.0 

2.5 


Prop. Pts. 


74 < 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


43 


16° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.44 034 


9.45 750 


10.54 250 

9.98 284 


60 

1 

9.44 078 


9.45 797 

47 

10.54 203 

9.98 281 

3 

59 

2 

9.44 122 

44 

9.45 845 

48 

10.54 155 

9.98 277 

4 

58 

3 

9.44 166 

44 

9.45 892 

47 

10.54 108 

9.98 273 

4 

57 

4 

9.44 210 

44 

43 

9.45 940 

48 

47 

10.54 060 

9.98 270 

3 

56 

5 

9.44 253 

9.45 987 

10.54 013 

9.98 266 

4 

55 

6 

9.44 297 

44 

9.46 035 

48 

10.53 965 

9.98 262 

4 

54 

7 

9.44 341 

44 

9.46 082 

47 

10.53 918 

9.98 259 

3 

53 

8 

9.44 385 

44 

9.46 130 

48 

10.53 870 

9.98 255 

4 

52 

9 

9.44 428 

43 

44 

9.46 177 

47 

47 

10.53 823 

9.98 251 

4 

51 

10 

9.44 472 

9.46 224 

10.53 776 

9.98 248 

3 

50 

11 

9.44 516 

44 

9.46 271 

47 

10.53 729 

9.98 244 

4 

49 

12 

9.44 559 

43 

9.46 319 

48 

10.53 681 

9.98 240 

4 

48 

13 

9.44 602 

43 

9.46 366 

47 

10.53 634 

9.98 237 

3 

47 

14 

9.44 646 

44 

43 

9.46 413 

47 

47 

10.53 587 

9.98 233 

4 

46 

15 

9.44 689 

9.46 460 

10.53 540 

9.98 229 

4 

45 

16 

9.44 733 

44 

9.46 507 

47 

10.53 493 

9.98 226 

3 

44 

17 

9.44 776 

43 

9.46 554 

47 

10.53 446 

9.98 222 

4 

43 

18 

9.44 819 

43 

9.46 601 

47 

10.53 399 

9.98 218 

4 

42 

19 

9.44 862 

43 

43 

9.46 648 

47 

46 

10.53 352 

9.98 215 

3 

41 

20 

9.44 905 

9.46 694 

10.53 306 

9.98 211 

4 

40 

21 

9.44 948 

43 

9.46 741 

47 

10.53 259 

9.98 207 

4 

39 

22 

9.44 992 

44 

9.46 788 

47 

10.53 212 

9.98 204 

3 

38 

23 

9.45 035 

43 

9.46 835 

47 

10.53 165 

9.98 200 

4 

37 

24 

9.45 077 

42 

43 

9.46 881 

46 

47 

10.53 119 

9.98 196 

4 

36 

25 

9.45 120 

9.46 928 

10.53 072 

9.98 192 

4 

35 

26 

9.45 163 

43 

9.46 975 

47 

10.53 025 

9.98 189 

3 

34 

27 

9.45 206 

43 

9.47 021 

46 

10.52 979 

9.98 185 

4 

33 

28 

9.45 249 

43 

9.47 068 

47 

10.52 932 

9.98 181 

4 

32 

29 

9.45 292 

43 

42 

9.47 114 

46 

46 

10.52 886 

9.98 177 

4 

Q 

31 

30 

9.45 344 

9.47 160 

10.52 840 

9.98 174 

o 

30 

31 

9.45 377 

43 

9.47 207 

47 

10.52 793 

9.98 170 

4 

29 

32 

9.45 419 

42 

9.47 253 

46 

10.52 747 

9.98 166 

4 

28 

33 

9.45 462 

43 

9.47 299 

46 

10.52 701 

9.98 162 

4 

27 

34 

9.45 504 

42 

43 

9.47 346 

47 

46 

10.52 654 

9.98 159 

3 

A 

26 

35 

9.45 547 

9.47 392 

10.52 608 

9.98 155 


25 

36 

9.45 589 

42 

9.47 438 

46 

10.52 562 

9.98 151 

4 

24 

37 

9.45 632 

43 

9.47 484 

46 

10.52 516 

9.98 147 

4 

23 

38 

9.45 674 

42 

9.47 530 

46 

10.52 470 

9.98 144 

3 

22 

39 

9.45 716 

42 

42 

9.47 576 

46 

46 

10.52 424 

9.98 140 

4 

A 

21 

40 

9.45 758 

9.47 622 

10.52 378 

9.98 136 


20 

41 

9.45 801 

43 

9.47 668 

46 

10.52 332 

9.98 132 

4 

19 

42 

9.45 843 

42 

9.47 714 

46 

10.52 286 

9.98 129 

3 

18 

43 

9.45 885 

42 

9.47 760 

46 

10.52 240 

9.98 125 

4 

17 

44 

9.45 927 

42 

42 

9.47 806 

46 

Ad 

10.52 194 

9.98 121 

4 

A 

16 

45 

9.45 969 

9.47 852 


10.52 148 

9.98 117 


15 

46 

9.46 011 

42 

9.47 897 

45 

10.52 103 

9.98 113 

4 

14 

47 

9.46 053 

42 

9.47 943 

46 

10.52 057 

9.98 110 

3 

13 

48 

9.46 095 

42 

9.47 989 

46 

10.52 011 

9.98 106 

4 

12 

49 

9.46 136 

41 

A O 

9.48 035 

46 

A r. 

10.51 965 

9.98 102 

4 

4. 

11 

50 

9.46 178 


9.48 080 

40 

10.51 920 

9.98 098 


10 

51 

9.46 220 

42 

9.48 126 

46 

10.51 874 

9.98 094 

4 

9 

52 

9.46 262 

42 

9.48 171 

45 

10.51 829 

9.98 090 

4 

8 

53 

9.46 303 

41 

9.48 217 

46 

10.51 783 

9.98 087 

3 

7 

54 

9.46 345 

42 

41 

9.48 262 

45 

AZ 

10.51 738 

9.98 083 

4 

4 

6 

55 

9.46 386 


9.48 307 

40 

10.51 693 

9.98 079 


5 

56 

9.46 428 

42 

9.48 353 

46 

10.51 647 

9.98 075 

4 

4 

57 

9.46 469 

41 

9.48 398 

45 

10.51 602 

9.98 071 

4 

3 

58 

9.46 511 

42 

9.48 443 

45 

10.51 557 

9.98 067 

4 

2 

59 

9.46 552 

41 

AO 

9.48 489 

46 

A d 

10.51 511 

9.98 063 

4 

3 

1 

60 

9.46 594 


9.48 534 

40 

10.51 466 

9.98 060 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

/ 


Prop. Pts. 


" 

48 

47 

46 

6 

4.8 

4.7 

4.6 

7 

5.6 

5.5 

5.4 

8 

6.4 

6.3 

6.1 

9 

7.2 

7.0 

6.9 

10 

8.0 

7.8 

7.7 

20 

16.0 

15.7 

15.3 

30 

24.0 

23.5 

23.0 

40 

32.0 

31.3 

30.7 

50 

40.0 

39.2 

38.3 


" 

45 

44 

6 

4.5 

4.4 

7 

5.3 

5.1 

8 

6.0 

5.9 

9 

6.8 

6.6 

10 

7.5 

7.3 

20 

15.0 

14.7 

30 

22.5 

22.0 

40 

30.0 

29.3 

50 

37.5 

36.7 


43 

4.3 
5.0 
5.7 

6.4 
7.2 

14.3 

21.5 

28.7 

35.8 


42 

4.2 
4.9 
5.6 

6.3 
7.0 

14.0 

21.0 

28.0 

35.0 


41 

4.1 

4.8 

5.5 

6.2 

6.8 
13.7 

20.5 
27.3 
34.2 


0.3 

0.4 

0.4 

0.5 

0.5 

1.0 

1.5 
2.0 

2.5 


Prop. Pts. 


73 













































































44 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


17 c 


/ 

L Sin 

0 

9.46 594 

1 

9.46 635 

2 

9.46 676 

3 

9.46 717 

4 

9.46 758 

5 

9.46 800 

6 

9.46 841 

7 

9.46 882 

8 

9.46 923 

9 

9.46 964 

10 

9.47 005 

11 

9.47 045 

12 

9.47 086 

13 

9.47 127 

14 

9.47 168 

15 

9.47 209 

16 

9.47 249 

17 

9.47 290 

18 

9.47 330 

19 

9.47 371 

20 

9.47 411 

21 

9.47 452 

22 

9.47 492 

23 

9.47 533 

24 

9.47 573 

25 

9.47 613 

26 

9.47 654 

27 

9.47 694 

28 

9.47 734 

29 

9.47 774 

30 

9.47 814 

31 

9.47 854 

32 

9.47 894 

33 

9.47 934 

34 

9.47 974 

35 

9.48 014 

36 

9.48 054 

37 

9.48 094 

38 

9.48 133 

39 

9.48 173 

40 

9.48 213 

41 

9.48 252 

42 

9.48 292 

43 

9.48 332 

44 

9.48 371 

45 

9.48 411 

46 

9.48 450 

47 

9.48 490 

48 

9.48 529 

49 

9.48 568 

50 

9.48 607 

51 

9.48 647 

52 

9.48 686 

53 

9.48 725 

54 

9.48 764 

55 

9.48 803 

56 

9.48 842 

57 

9.48 881 

58 

9.48 920 

59 

9.48 959 

60 

9.48 998 


L Cos 


L Tan 

c d 

L Cot 

L Cos 

d 


9.48 534 


10.51 466 

9.98 060 


60 

9.48 579 

45 

10.51 421 

9.98 056 

4 

59 

9.48 624 

45 

10.51 376 

9.98 052 

4 

58 

9.48 669 

45 

10.51 331 

9.98 048 

4 

57 

9.48 714 

45 

10.51 286 

9.98 044 

4 

A 

56 

9.48 759 


10.51 241 

9.98 040 

tr 

55 

9.48 804 

45 

10.51 196 

9.98 036 

4 

54 

9.48 849 

45 

10.51 151 

9.98 032 

4 

53 

9.48 894 

45 

10.51 106 

9.98 029 

3 

52 

9.48 939 

45 

45 

10.51 061 

9.98 025 

4 

4 

51 

9.48 984 

10.51 016 

9.98 021 


50 

9.49 029 

45 

10.50 971 

9.98 017 

4 

49 

9.49 073 

44 

10.50 927 

9.98 013 

4 

48 

9.49 118 

45 

10.50 882 

9.98 009 

4 

47 

9.49 163 

45 

44 

10.50 837 

9.98 005 

4 

4 

46 

9.49 207 

10.50 793 

9.98 001 


45 

9.49 252 

45 

10.50 748 

9.97 997 

4 

44 

9.49 296 

44 

10.50 704 

9.97 993 

4 

43 

9.49 341 

45 

10.50 659 

9.97 989 

4 

42 

9.49 385 

44 

45 

10.50 615 

9.97 986 

3 

4 

41 

9.49 430 

10.50 570 

9.97 982 


40 

9.49 474 

44 

10.50 526 

9.97 978 

4 

39 

9.49 519 

45 

10.50 481 

9.97 974 

4 

38 

9.49 563 

44 

10.50 437 

9.97 970 

4 

37 

9.49 607 

44 

45 

10.50 393 

9.97 966 

4 

4 

36 

9.49 652 


10.50 348 

9.97 962 


35 

9.49 696 

44 

10.50 304 

9.97 958 

4 

34 

9.49 740 

44 

10.50 260 

9.97 954 

4 

33 

9.49 784 

44 

10.50 216 

9.97 950 

4 

32 

9.49 828 

44 

44 

10.50 172 

9.97 946 

4 

4 

31 

9.49 872 

10.50 128 

9.97 942 


30 

9.49 916 

44 

10.50 084 

9.97 938 

4 

29 

9.49 960 

44 

10.50 040 

9.97 934 

4 

28 

9.50 004 

44 

10.49 996 

9.97 930 

4 

27 

9.50 048 

44 

44 

10.49 952 

9.97 926 

4 

4 

26 

9.50 092 

10.49 908 

9.97 922 


25 

9.50 136 

44 

10.49 864 

9.97 918 

4 

24 

9.50 180 

44 

10.49 820 

9.97 914 

4 

23 

9.50 223 

43 

10.49 777 

9.97 910 

4 

22 

9.50 267 

44 

44 

10.49 733 

9.97 906 

4 

4 

21 

9.50 311 

10.49 689 

9.97 902 


20 

9.50 355 

44 

10.49 645 

9.97 898 

4 

19 

9.50 398 

43 

10.49 602 

9.97 894 

4 

18 

9.50 442 

44 

10.49 558 

9.97 890 

4 

17 

9.50 485 

43 

44 

10.49 515 

9.97 886 

4 

4 

16 

9.50 529 

10.49 471 

9.97 882 


15 

9.50 572 

43 

10.49 428 

9.97 878 

4 

14 

9.50 616 

44 

10.49 384 

9.97 874 

4 

13 

9.50 659 

43 

10.49 341 

9.97 870 

4 

12 

9.50 703 

44 

10.49 297 

9.97 866 

4 

5 

11 

9.50 746 

TfcO 

10.49 254 

9.97 861 


10 

9.50 789 

43 

10.49 211 

9.97 857 

4 

9 

9.50 833 

44 

10.49 167 

9.97 853 

4 

8 

9.50 876 

43 

10.49 124 

9.97 849 

4 

7 

9.50 919 

43 

43 

10.49 081 

9.97 845 

4 

4 

6 

9.50 962 

10.49 038 

9.97 841 


5 

9.51 005 

43 

10.48 995 

9.97 837 

4 

4 

9.51 048 

43 

10.48 952 

9.97 833 

4 

3 

9.51 092 

44 

10.48 908 

9.97 829 

4 

2 

9.51 135 

43 
• 43 

10.48 865 

9.97 825 

4 

4 

1 

9.51 178 

10.48 822 

9.97 821 


0 

L Cot 

c d 

L Tan 

L Sin 

d 

f 


41 

41 

41 

41 

42 
41 
41 
41 
41 
41 

40 

41 
41 
41 
41 

40 

41 

40 

41 

40 

41 

40 

41 
40 

40 

41 
40 
40 
40 
40 
40 
40 
40 
40 
40' 
40 
40 

39 

40 
40 

39 

40 
40 

39 

40 

39 

40 
39 


Prop. Pts. 


>> 

45 

44 

43 

6 

4.5 

4.4 

4.3 

7 

5.3 

5.1 

5.0 

8 

6.0 

5.9 

5.7 

9 

6.8 

6.6 

6.4 

10 

7.5 

7.3 

7.2 

20 

15.0 

14.7 

14.3 

30 

22.5 

22.0 

21.5 

40 

30.0 

29.3 

28.7 

50 

37.5 

36.7 

35.8 


42 

4.2 
4.9 
5.6 

6.3 
7.0 

14.0 

21.0 

28.0 

35.0 


40 

4.0 

4.7 
5.3 
6.0 

6.7 

13.3 
20.0 
26.7 

33.3 


41 

4.1 

4.8 
5.5 

6.2 

6.8 
13.7 
20.5 
27.3 
34.2 


39 


4.6 

5.2 

5.9 

6.5 

13.0 

19.5 
26.0 

32.5 


5 

4 

3 

0.5 

0.4 

0.3 

0.6 

0.5 

0.4 

0.7 

0.5 

0.4 

0.8 

0.6 

0.5 

0.8 

0.7 

0.5 

1.7 

1.3 

1.0 

2.5 

2.0 

1.5 

3.3 

2.7 

2.0 

4.2 

3.3 

2.5 


Prop. Pts. 


72 ‘ 




























































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 

18 ° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

1 

2 

3 

4 

9.48 998 

9.49 037 
9.49 076 
9.49 115 
9.49 153 

39 

39 

39 

38 

39 

39 

38 

39 
39 

38 

39 
38 

38 

39 
38 

38 

39 
38 
38 
38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

38 

37 

38 
38 

37 

38 

38 

37 

38 
37 

37 

38 
37 

37 

38 
37 

37 

37 

37 

37 

38 

37 

37 

37 

36 

37 

37 

37 

37 

36 

37 

9.51 178 
9.51 221 
9.51 264 
9.51 306 
9.51 349 

43 

43 

42 

43 
43 

43 

43 

42 

43 
43 

42 

43 
43 

42 

43 

42 

42 

43 

42 

43 

42 

42 

42 

43 
42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

41 

42 
42 
42 
42 

41 

42 

41 

42 
42 

41 

42 
41 

41 

42 

41 

42 
41 
41 

41 

42 

41 

41 

41 

41 

41 

10.48 822 
10.48 779 
10.48 736 
10.48 694 
10.48 651 

9.97 821 
9.97 817 
9.97 812 
9.97 808 
9.97 804 

4 

5 

4 

4 

4 

4 

4 

4 

4 

5 

4 

4 

4 

4 

4 

5 

4 

4 

4 

4 

4 

5 

4 

4 

4 

4 

5 

4 

4 

4 

5 

4 

4 

4 

5 

4 

4 

4 

5 

4 

4 

4 

5 

4 

4 

4 

5 

4 

4 

5 

4 

4 

5 

4 

4 

5 

4 

4 

5 

4 

60 

59 

58 

57 

56 

5 

6 

7 

8 

9 

9.49 192 
9.49 231 
9.49 269 
9.49 308 
9.49 347 

9.51 392 
9.51 435 
9.51 478 
9.51 520 
9.51 563 

10.48 608 

10.48 565 
10.48 522 
10.48 480 
10.48 437 

9.97 800 
9.97 796 
9.97 792 
9.97 788 
9.97 784 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

10 

11 

12 

13 

14 

9.49 385 
9.49 424 
9.49 462 
9.49 500 
9.49 539 

9.51 606 
9.51 648 
9.51 691 
9.51 734 
9.51 776 

10.48 394 

10.48 352 
10.48 309 
10.48 266 
1048 224 

9.97 779 
9.97 775 
9.97 771 
9.97 767 
9.97 763 

15 

16 

17 

18 
19 

9.49 577 
9.49 615 
9.49 654 
9.49 692 
9.49 730 

9.51 819 
9.51 861 
9.51 903 
9.51 946 
9.51 988 

10.48 181 

10.48 139 
10.48 097 
10.48 054 
10.48 012 

9.97 759 
9.97 754 
9.97 750 
9.97 746 
9.97 742 

20 

21 

22 

23 

24 

9.49 768 
9.49 806 
9.49 844 
9.49 882 
9.49 920 

9.52 031 
9.52 073 
9.52 115 
9.52 157 
9.52 200 

10.47 969 
10.47 927 
10.47 885 
10.47 843 
10.47 800 

9.97 738 
9.97 734 
9.97 729 
9.97 725 
9.97 721 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.49 958 

9.49 996 

9.50 034 
9.50 072 
9.50 110 

9.52 242 
9.52 284 
9.52 326 
9.52 368 
9.52 410 

10.47 758 
10.47 716 
10.47 674 
10.47 632 
10.47 590 

9.97 717 
9.97 713 
9.97 708 
9.97 704 
9.97 700 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.50 148 
9.50 185 
9.50 223 
9.50 261 
9.50 298 

9.52 452 
9.52 494 
9.52 536 
9.52 578 
9.52 620 

10.47 548 
10.47 506 
10.47 464 
10.47 422 
10.47 380 

9.97 696 
9.97 691 
9.97 687 
9.97 683 
9.97 679 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.50 336 
9.50 374 
9.50 411 
9.50 449 
9.50 486 

9.52 661 
9.52 703 
9.52 745 
9.52 787 
9.52 829 

10.47 339 
10.47 297 
10.47 255 
10.47 213 
10.47 171 

9.97 674 
9.97 670 
9.97 666 
9.97 662 
9.97 657 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.50 523 
9.50 561 
9.50 598 
9.50 635 
9.50 673 

9.52 870 
9.52 912 
9.52 953 

9.52 995 

9.53 037 

10.47 130 
10.47 088 
10.47 047 
10.47 005 
10.46 963 

9.97 653 
9.97 649 
9.97 645 
9.97 640 
9.97 636 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.50 710 
9.50 747 
9.50 784 
9.50 821 
9.50 858 

9.53 078 
9.53 120 
9.53 161 
9.53 202 
9.53 244 

10.46 922 
10.46 880 
10.46 839 
10.46 798 
10.46 756 

9.97 632 
9.97 628 
9.97 623 
9.97 619 
9.97 615 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.50 896 
9.50 933 

9.50 970 

9.51 007 
9.51 043 

9.53 285 
9.53 327 
9.53 368 
9.53 409 
9.53 450 

10.46 715 
10.46 673 
10.46 632 
10.46 591 
10.46 550 

9.97 610 
9.97 606 
9.97 602 
9.97 597 
9.97 593 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.51 080 
9.51 117 
9.51 154 
9.51 191 
9.51 227 

9.53 492 
9.53 533 
9.53 574 
9.53 615 
9.53 656 

10.46 508 
10.46 467 
10.46 426 
10.46 385 
10.46 344 

9.97 589 
9.97 584 
9.97 580 
9.97 576 
9.97 571 

5 

4 

3 

2 

1 

60 

9.51 264 

9.53 697 

10.46 303 

9.97 567 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 


Prop. Pts. 


" 

43 

42 

41 

6 

4.3 

4.2 

4.1 

7 

5.0 

4.9 

4.8 

8 

5.7 

5.6 

5.5 

9 

. 6.4 

6.3 

6.2 

10 

7.2 

7.0 

6.8 

20 

14.3 

14.0 

13.7 

30 

21.5 

21.0 

20.5 

40 

28.7 

28.0 

27.3 

50 

35.8 

35.0 

34.2 


" 

39 

38 

37 

6 

3.9 

3.8 

3.7 

7 

4.6 

4.4 

4.3 

8 

5.2 

5.1 

4.9 

9 

5.9 

5.7 

5.6 

10 

6.5 

6.3 

6.2 

20 

13.0 

12.7 

12.3 

30 

19.5 

19.0 

18.5 

40 

26.0 

25.3 

24.7 

50 

32.5 

31.7 

30.8 


" 

36 

5 

4 

6 

3.6 

0.5 

0.4 

7 

4.2 

0.6 

0.5 

8 

4.8 

0.7 

0.5 

9 

5.4 

0.8 

0.6 

10 

6.0 

0.8 

0.7 

20 

12.0 

1.7 

1.3 

30 

18.0 

2.5 

2.0 

40 

24.0 

3.3 

2.7 

50 

30.0 

4.2 

3.3 


Prop. Pts. 


IV 

















































































46 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 

19 ° 


VI 


' 

L Sin 

d 

L T 3.11 

cd 

L Cot 

L Cos 

o 

9.51 264 


9.53 697 

41 

10.46 303 

9.97 567 

1 

9 51 301 

37 

9.53 738 

10.46 262 

9.97 563 

2 

9 51 338 

37 

9.53 779 

41 

10.46 221 

9.97 558 

3 

9 51 374 

36 

9.53 820 

41 

10.46 180 

9.97 554 

4 

9.51 411 

37 

9.53 861 

41 

41 

10.46 139 

9.97 550 

5 

9 51 447 


9.53 902 

41 

10.46 098 

9.97 545 

6 

9.51 484 

37 

9.53 943 

10.46 057 

9.97 541 

7 

9 51 520 

36 

9.53 984 

41 

10.46 016 

9.97 536 

8 

9 51 557 

37 

9.54 025 

41 

10.45 975 

9.97 532 

9 

9.51 593 

36 

9.54 065 

40 

41 

10.45 935 

9.97 528 

10 

9.51 629 


9.54 106 

41 

10.45 894 

9.97 523 

11 

9 51 666 ■ 

37 

9.54 147 

10.45 853 

9.97 519 

12 

9 51 702 

36 

9.54 187 

40 

10.45 813 

9.97 515 

13 

9.51 738 

36 

9.54 228 

41 

10.45 772 

9.97 510 

14 

9.51 774 

36 

9.54 269 

41 

10.45 731 

9.97 506 

15 

9.51 811 


9.54 309 

41 

10.45 691 

9.97 501 

16 

9.51 847 

36 

9.54 350 

10.45 650 

9.97 497 

17 

9.51 883 

36 

9.54 390 

40 

10.45 610 

9.97 492 

18 

9.51 919 

36 

9.54 431 

41 

10.45 569 

9.97 488 

19 

9.51 955 

36 

9.54 471 

40 

10.45 529 

9.97 484 

20 

9.51 991 


9.54 512 

41 

10.45 488 

9.97 479 

21 

9.52 027 

36 

9.54 552 

40 

10.45 448 

9.97 475 

22 

9.52 063 

36 

9.54 593 

41 

10.45 407 

9.97 470 

23 

9.52 099 

36 

9.54 633 

40 

10.45 367 

9.97 466 

24 

9.52 135 

36 

9.54 673 

40 

10.45 327 

9.97 461 

25 

9.52 171 

36 

9.54 714 

40 

10.45 286 

9.97 457 

26 

9.52 207 

36 

9.54 754 

10.45 246 

9.97 453 

27 

9.52 242 

35 

9.54 794 

40 

10.45 206 

9.97 448 

28 

9.52 278 

36 

9.54 835 

41 

10.45 165 

9.97 444 

29 

9.52 314 

36 

9.54 875 

40 

10.45 125 

9.97 439 

30 

9.52 350 

36 

9.54 915 

40 

10.45 085 

9.97 435 

31 

9.52 385 

35 

9.54 955 

10.45 045 

9.97 430 

32 

9.52 421 

36 

9.54 995 

40 

10.45 005 

9.97 426 

33 

9.52 456 

35 

9.55 035 

40 

10.44 965 

9.97 421 

34 

9.52 492 

36 

9.55 075 

40 

10.44 925 

9.97 417 

35 

9.52 527 

35 

9.55 115 

40 

10.44 885 

9.97 412 

36 

9.52 563 

36 

9.55 155 

10.44 845 

9.97 408 

37 

9.52 598 

35 

9.55 195 

40 

10.44 805 

9.97 403 

38 

9.52 634 

36 

9.55 235 

40 

10.44 765 

9.97 399 

39 

9.52 669 

35 

9.55 275 

40 

40 

10.44 725 

9.97 394 

40 

9.52 705 

36 

9.55 315 


10.44 685 

9.97 390 

41 

9.52 740 

35 

9.55 355 

40 

10.44 645 

9.97 385 

42 

9.52 775 

35 

9.55 395 

40 

10.44 605 

9.97 381 

43 

9.52 811 

36 

9.55 434 

39 

10.44 566 

9.97 376 

44 

9.52 846 

35 

35 

9.55 474 

40 

40 

10.44 526 

9.97 372 

45 

9.52 881 

9.55 514 

10.44 486 

9.97 367 

46 

9.52 916 

35 

9.55 554 

40 

10.44 446 

9.97 363 

47 

9.52 951 

35 

9.55 593 

39 

10.44 407 

9.97 358 

48 

9.52 986 

35 

9.55 633 

40 

10.44 367 

9.97 353 

49 

9.53 021 

35 
• 35 

9.55 673 

40 

10.44 327 

9.97 349 

50 

9.53 056 

9.55 712 

40 

10.44 288 

9.97 344 

51 

9.53 092 

36 

9.55 752 

10.44 248 

9.97 340 

52 

9.53 126 

34 

9.55 791 

39 

10.44 209 

9.97 335 

53 

9.53 161 

35 

9.55 831 

40 

10.44 169 

9.97 331 

54 

9.53 196 

35 
- 35 

9.55 870 

39 

40 

10.44 130 

9.97 326 

55 

9.53 231 

9.55 910 


10.44 090 

9.97 322 

56 

9.53 266 

35 

9.55 949 

39 

10.44 051 

9.97 317 

57 

9.53 301 

35 

9.55 989 

40 

10.44 011 

9.97 312 

58 

9.53 336 

35 

9.56 028 

39 

10.43 972 

9.97 308 

59 

9.53 370 

34 

9.56 067 

39 
- 40 

10.43 933 

9.97 303 

60 

9.53 405 


9.56 107 


10.43 893 

9.97 299 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 


Prop. Pts. 


" 

41 

40 

6 

4.1 

4.0 

7 

4.8 

4.7 

8 

5.5 

5.3 

9 

6.2 

6.0 

10 

6.8 

6.7 

20 

13.7 

13.3 

30 

20.5 

20.0 

40 

27.3 

26.7 

50 

34.2 

33.3 


3.9 
4.6 
5.2 

5.9 
6.5 

13.0 

19.5 
26.0 

32.5 


" 

37 

36 

35 

6 

3.7 

3.6 

3.5 

7 

4.3 

4.2 

4.1 

8 

4.9 

4.8 

4.7 

9 

5.6 

5.4 

5.3 

10 

6.2 

6.0 

5.8 

20 

12.3 

12.0 

11.7 

30 

18.5 

18.0 

17.5 

40 

24.7 

24.0 

23.3 

50 

30.8 

30.0 

29.2 


" 

34 

5 

4 

6 

3.4 

0.5 

0.4 

7 

4.0 

0.6 

0.5 

8 

4.5 

0.7 

0.5 

9 

5.1 

0.8 

0.6 

10 

5.7 

0.8 

0.7 

20 

11.3 

1.7 

1.3 

30 

17.0 

2.5 

2.0 

40 

22.7 

3.3 

2.7 

50 

28.3 

4.2 

3.3 


Prop. Pts. 


70 ° 



































































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


47 


20 c 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.53 405 


9.56 107 


10.43 893 

9.97 299 


60 

1 

9.53 440 

35 

9.56 146 

39 

10.43 854 

9.97 294 

5 

59 

2 

9.53 475 

35 

9.56 185 

39 

10.43 815 

9.97 289 

5 

58 

3 

9.53 509 

34 

9.56 224 

39 

10.43 776 

9.97 285 

4 

57 

4 

9.53 544 

35 

9.56 264 

40 

10.43 736 

9.97 280 

5 

56 



34 


39 





5 

9.53 578 

9.56 303 

10.43 697 

9.97 276 


55 

6 

9.53 613 

35 

9.56 342 

39 

10.43 658 

9.97 271 

5 

54 

7 

9.53 647 

34 

9.56 381 

39 

10.43 619 

9.97 266 

5 

53 

8 

9.53 682 

35 

9.56 420 

39 

10.43 580 

9.97 262 

4 

52 

9 

9.53 716 

34 

9.56 459 

39 

39 

10.43 541 

9.97 257 

5 

PL 

51 

10 

9.53 751 


9.56 498 

10.43 502 

9.97 252 


50 

11 

9.53 785 

34 

9.56 537 

39 

10.43 463 

9.97 248 

4 

49 

12 

9.53 819 

34 

9.56 576 

39 

10.43 424 

9.97 243 

5 

48 

13 

9.53 854 

35 

9.56 615 

39 

10.43 385 

9.97 238 

5 

47 

14 

9.53 888 

34 

9.56 654 

39 

39 

10.43 346 

9 97 234 

4 

5 

46 

15 

9.53 922 


9.56 693 

10.43 307 

9.97 229 


45 

16 

9.53 957 

35 

9.56 732 

39 

10.43 268 

9.97 224 

5 

44 

17 

9.53 991 

34 

9.56 771 

39 

10.43 229 

9.97 220 

4 

43 

18 

9.54 025 

34 

9.56 810 

39 

10.43 190 

9.97 215 

5 

42 

19 

9.54 059 

34 

9.56 849 

39 

90 

10.43 151 

9.97 210 

5 

4 

41 

20 

9.54 093 


9.56 887 

o o 

10.43 113 

9.97 206 


40 

21 

9.54 127 

34 

9.56 926 

39 

10.43 074 

9.97 201 

5 

39 

22 

9.54 161 

34 

9.56 965 

39 

10.43 035 

9.97 196 

5 

38 

23 

9.54 195 

34 

9.57 004 

39 

10.42 996 

9.97 192 

4 

37 

24 

9.54 229 

34 

9.57 042 

38 

39 

10.42 958 

9.97 187 

5 

5 

36 

25 

9.54 263 

04 

9.57 081 

10.42 919 

9.97 182 


35 

26 

9.54 297 

34 

9.57 120 

39 

10.42 880 

9.97 178 

4 

34 

27 

9.54 331 

34 

9.57 158 

38 

10.42 842 

9.97 173 

5 

33 

28 

9.54 365 

34 

9.57 197 

39 

10.42 803 

9.97 168 

5 

32 

29 

9.54 399 

34 

HA 

9.57 235 

38 

QQ 

10.42 765 

9.97 163 

5 

4 

31 

30 

9.54 433 


9.57 274 

o y 

10.42 726 

9.97 159 


30 

31 

9.54 466 

33 

9.57 312 

38 

10.42 688 

9.97 154 

5 

29 

32 

9.54 500 

34 

9.57 351 

39 

10.42 649 

9.97 149 

5 

28 

33 

9.54 534 

34 

9.57 389 

38 

10.42 611 

9.97 145 

4 

27 

34 

9.54 567 

33 

QA 

9.57 428 

39 

38 

10.42 572 

9.97 140 

5 

5 

26 

35 

9.54 601 

o4 

9.57 466 


10.42 534 

9.97 135 


25 

36 

9.54 635 

34 

9.57 504 

38 

10.42 496 

9.97 130 

5 

24 

37 

9.54 668 

33 

9.57 543 

39 

10.42 457 

9.97 126 

4 

23 

38 

9.54 702 

34 

9.57 581 

38 

10.42 419 

9.97 121 

5 

22 

39 

9.54 735 

33 

Q A 

9.57 619 

38 

•JQ 

10.42 381 

9.97 116 

5 

5 

21 

40 

9.54 769 

04 

9.57 658 

oy 

10.42 342 

9.97 111 

A 

20 

41 

9.54 802 

33 

9.57 696 

38 

10.42 304 

9.97 107 

4 

19 

42 

9.54 836 

34 

9.57 734 

38 

10.42 266 

9.97 102 

5 

18 

43 

9.54 869 

33 

9.57 772 

38 

10.42 228 

9.97 097 

5 

17 

44 

9.54 903 

34 

9.57 810 

38 

QQ 

10.42 190 

9.97 092 

5 

5 

16 

45 

9.54 936 

33 

9.57 849 

oy 

10.42 151 

9.97 087 

A 

15 

46 

9.54 969 

33 

9.57 887 

38 

10.42 113 

9.97 083 

4 

14 

47 

9.55 003 

34 

9.57 925 

38 

10.42 075 

9.97 078 

5 

r 

13 

48 

9.55 036 

33 

9.57 963 

38 

10.42 037 

9.97 073 

O 

12 

49 

9.55 069 

33 

9.58 001 

38 

QO 

10.41 999 

9.97 068 

5 

5 

11 

50 

9.55 102 

33 

9.58 039 

oo 

10.41 961 

9.97 063 

A 

10 

51 

9.55 136 

34 

9.58 077 

38 

10.41 923 

9.97 059 

4 

rr 

9 

52 

9.55 169 

33 

9.58 115 

38 

10.41 885 

9.97 054 

5 

8 

53 

9.55 202 

33 

9.58 153 

38 

10.41 847 

9.97 049 

5 

7 

54 

9.55 235 

33 

9.58 191 

38 

QC 

10.41 809 

9.97 044 

5 

5 

6 

55 

9.55 268 

33 

9.58 229 

Oo 

10.41 771 

9.97 039 

A 

5 

56 

9.55 301 

33 

9.58 267 

38 

10.41 733 

9.97 035 

4 

r 

4 

57 

9.55 334 

33 

9.58 304 

37 

10.41 696 

9.97 030 

O 

c 

3 

58 

9.55 367 

33 

9.58 342 

38 

10.41 658 

9.97 025 

O 

2 

59 

9.55 400 

33 

9.58 380 

38 

QO 

10.41 620 

9.97 020 

o 

5 

1 

60 

9.55 433 

33 

9.58 418 

OO 

10.41 582 

9.97 015 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

f 


Prop. Pts. 


40 

4.0 

4.7 
5.3 
6.0 

6.7 

13.3 
20.0 
26.7 

33.3 


4.4 

5.1 

5.7 

6.3 

12.7 
19.0 
25.3 

31.7 


35 

3.5 

4.1 

4.7 
5.3 

5.8 
11.7 
17.5 
23.3 
29.2 


39 

3.9 
4.6 
5.2 

5.9 
6.5 

13.0 

19.5 
26.0 

32.5 


37 

3.7 

4.3 

4.9 

5.6 

6.2 

12.3 

18.5 

24.7 

30.8 


34 

3.4 
4.0 

4.5 
5.1 
5.7 

11.3 
17.0 
22.7 

28.3 


" 

33 

5 

4 

6 

3.3 

0.5 

0.4 

7 

3.8 

0.6 

0.5 

8 

4.4 

0.7 

0.5 

9 

5.0 

0.8 

0.6 

10 

5.5 

0.8 

0.7 

20 

11.0 

1.7 

1.3 

30 

16.5 

2.5 

2.0 

40 

22.0 

3.3 

2.7 

50 

27.5 

4.2 

3.3 


Prop. Pts. 


69 ‘ 






















































































48 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


21 ° 


' 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

9.55 433 


9.58 418 


10.41 582 

9.97 015 


60 






1 

9.55 466 

33 

9.58 455 

37 

10.41 545 

9.97 010 

5 

59 






2 

9.55 499 

33 

9.58 493 

38 

10.41 507 

9.97 005 

5 

58 






3 

9.55 532 

33 

9.58 531 

38 

10.41 469 

9.97 001 

4 

57 






4 

9.55 564 

32 

9.58 569 

38 

10.41 431 

9.96 996 

5 

5 

56 






5 

9.55 597 


9.58 606 


10.41 394 

9.96 991 


55 






6 

9.55 630 

33 

9.58 644 

38 

10.41 356 

9.96 986 

5 

54 






7 

9.55 663 

33 

9.58 681 

37 

10.41 319 

9.96 981 

5 

53 

n 

j 

18 

37 

36 

8 

9.55 695 

32 

9.58 719 

38 

10.41 281 

9.96 876 

5 

52 






9 

9.55 728 

33 

9.58 757 

38 

10.41 243 

9.96 971 

5 

5 

51 

6 

7 

3.8 
4 4 

3.7 
4 3 

3.6 

4 2 

10 

9.55 761 


9.58 794 


10.41 206 

9.96 966 


50 

4 

8 

5il 

4.9 

4^8 

11 

9.55 793 

32 

9.58 832 

38 

10.41 168 

9.96 962 

4 

49 

9 

5.7 

5.6 

5.4 

12 

9.55 826 

33 

9.58 869 

37 

10.41 131 

9.96 957 

5 

48 

10 

6.3 

6.2 

6.0 

13 

9.55 858 

32 

9.58 907 

38 

10.41 093 

9.96 952 

5 

47 

20 

12.7 

12.3 

12.0 

14 

9.55 891 

33 

9.58 944 

37 

37 

10.41 056 

9.96 947 

5 

5 

46 

30 

40 

19.0 

25.3 

18.5 

24.7 

18.0 

24.0 

15 

9.55 923 


9.58 981 

10.41 019 

9.96 942 


45 

50 

31.7 

30.8 

30.0 

16 

9.55 956 

33 

9.59 019 

38 

10.40 981 

9.96 937 

5 

44 






17 

9.55 988 

32 

9.59 056 

37 

10.40 944 

9.96 932 

5 

43 






18 

9.56 021 

33 

9.59 094 

38 

10.40 906 

9.96 927 

5 

42 






19 

9.56 053 

32 

9.59 131 

37 

10.40 869 

9.96 922 

5 

5 

41 






20 

9.56 085 


9.59 168 


10.40 832 

9.96 917 


40 






21 

9.56 118 

33 

9.59 205 

37 

10.40 795 

9.96 912 

5 

39 






22 

9.56 150 

32 

9.59 243 

38 

10.40 757 

9.96 907 

5 

38 






23 

9.56 182 

32 

9.59 280 

37 

10.40 720 

9.96 903 

4 

37 






24 

9.56 215 

33 

QO 

9.59 317 

37 

Q7 

10.40 683 

9.96 898 

5 

5 

36 






25 

9.56 247 

OZ 

9.59 354 

o l 

10.40 646 

9.96 893 


35 






26 

9.56 279 

32 

9.59 391 

37 

10.40 609 

9.96 888 

5 

34 

>> 

33 

32 

31 

27 

9.56 311 

32 

9.59 429 

38 

10.40 571 

9.96 883 

5 

33 






28 

9.56 343 

32 

9.59 466 

37 

10.40 534 

9.96 878 

5 

32 

6 


3.3 

3.2 

q n 

3.1 

Q 

29 

9.56 375 

32 

QQ 

9.59 503 

37 

Q7 

10.40 497 

9.96 873 

5 

5 

31 

7 

8 

0.3 

4.4 

o. / 
4.3 

0.0 

4.1 

30 

9.56 408 

oo 

9.59 540 

o ( 

10.40 460 

9.96 868 


30 

9 

5.0 

4.8 

4.6 

31 

9.56 440 

32 

9.59 577 

37 

10.40 423 

9.96 863 

5 

29 

10 

5.5 

5.3 

5.2 

32 

9.56 472 

32 

9.59 614 

37 

10.40 386 

9.96 858 

5 

28 

20 

30 

11.0 
16 5 

10.7 

16.0 

10.3 

15.5 

33 

9.56 504 

32 

9.59 651 

37 

10.40 349 

9.96 853 

5 

27 

40 

22.0 

21^3 

20^7 

34 

9.56 536 

32 

9.59 688 

37 

Q7 

10.40 312 

9.96 848 

5 

5 

26 

50 

27.5 

26.7 

25.8 

35 

9.56 568 

32 

9.59 725 

o t 

10.40 275 

9.96 843 


25 






36 

9.56 599 

31 

9.59 762 

37 

10.40 238 

9.96 838 

5 

24 






37 

9.56 631 

32 

9.59 799 

37 

10.40 201 

9.96 833 

5 

23 






38 

9.56 663 

32 

9.59 835 

36 

10.40 165 

9.96 828 

5 

22 






39 

9.56 695 

32 

QO 

9.59 872 

37 

Q7 

10.40 128 

9.96 823 

5 

K 

21 






40 

9.56 727 

oZ 

9.59 909 

O 4 

10.40 091 

9.96 818 

cl 

20 






41 

9.56 759 

32 

9.59 946 

37 

10.40 054 

9.96 813 

5 

19 






42 

9.56 790 

31 

9.59 983 

37 

10.40 017 

9.96 808 

5 

18 






43 

9.56 822 

32 

9.60 019 

36 

10.39 981 

9.96 803 

5 

17 






44 

9.56 854 

32 

QO 

9.60 056 

37 

10.39 944 

9.96 798 

5 

K 

16 






45 

9.56 886 

oZ 

9.60 093 

37 

10.39 907 

9.96 793 

cl 

15 



6 

5 

4 

46 

9.56 917 

31 

9.60 130 

37 

10.39 870 

9.96 788 

5 

14 






47 

9.56 949 

32 

9.60 166 

36 

10.39 834 

9.96 783 

5 

13 


6 

7 

0.6 
0 7 

0.5 
0 6 

0.4 

0 5 

48 

9.56 980 

31 

9.60 203 

37 

10.39 797 

9.96 778 

5 

12 


8 

0^8 

0^7 

0.5 

49 

9.57 012 

32 

QO 

9.60 240 

37 

QA 

10.39 760 

9.96 772 

6 

5 

11 


9 

0.9 

0.8 

0.6 

50 

9.57 044 

OZ 

9.60 276 

oO 

10.39 724 

9.96 767 


10 

1U 

90 

1.0 
9 n 

i 0.8 
i 1 7 

0.7 

1 3 

51 

9.57 075 

31 

9.60 313 

37 

10.39 687 

9.96 762 

5 

9 

30 

z.u 

3.0 

\ 2.5 

2.0 

52 

9.57 107 

32 

9.60 349 

36 

10.39 651 

9.96 757 

5 

8 

40 

4.0 

' 3.3 

2.7 

53 

9.57 138 

31 

9.60 386 

37 

10.39 614 

9.96 752 

5 

7 

50 

5.0 

i 4.2 

3.3 

54 

9.57 169 

31 

Q O 

9.60 422 

36 

Q7 

10.39 578 

9.96 747 

5 

K 

6 






55 

9.57 201 

OZ 

9.60 459 

ol 

10.39 541 

9.96 742 

O 

5 






56 

9.57 232 

31 

9.60 495 

36 

10.39 505 

9.96 737 

5 

4 






57 

9.57 264 

32 

9.60 532 

37 

10.39 468 

9.96 732 

5 

3 






58 

9.57 295 

31 

9.60 568 

36 

10.39 432 

9.96 727 

5 

2 






59 

9.57 326 

31 

32 

9.60 605 

37 

QA 

10.39 395 

9.96 722 

5 

r 

1 






60 

9.57 358 

9.60 641 

OO 

10.39 359 

9.96 717 

cl 

0 







L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


68 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


49 


22 ° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

9.57 358 


9.60 641 


10.39 359 

9.96 717 


60 







1 

9.57 389 

31 

9.60 677 

36 

10.39 323 

9.96 711 

6 

59 







2 

9.57 420 

31 

9.60 714 

37 

10.39 286 

9.96 706 

5 

58 







3 

9.57 451 

31 

9.60 750 

36 

10.39 250 

9.96 701 

5 

57 







4 

9.57 482 

31 

32 

9.60 786 

36 

10.39 214 

9.96 696 

5 

K 

56 







5 

9.57 514 

9.60 823 


10.39 177 

9.96 691 

o 

55 







6 

9.57 545 

31 

9.60 859 

36 

10.39 141 

9.96 686 

5 

54 







7 

9.57 576 

31 

9.60 895 

36 

10.39 105 

9.96 681 

5 

53 

// 


37 


36 

35 

8 

9.57 607 

31 

9.60 931 

36 

10.39 069 

9.96 676 

5 

52 







9 

9.57 638 

31 

31 

9.60 967 

36 

37 

10.39 033 

9.96 670 

6 

5 

51 

6 

7 


3.7 

A Q 


3.6 

A 9 

3.5 

10 

9.57 669 

9.61 004 

10.38 996 

9.96 665 


50 

/ 

8 


4.0 

4.9 


4.Z 

4.8 

4.7 

11 

9.57 700 

31 

9.61 040 

36 

10.38 960 

9.96 660 

5 

49 

9 


5.6 


5.4 

5.2 

12 

9.57 731 

31 

9.61 076 

36 

10.38 924 

9.96 655 

5 

48 

10 


6.2 


6.0 

5.8 

13 

9.57 762 

31 

9.61 112 

36 

10.38 888 

9.96 650 

5 

47 

20 

12.3 

12.0 

11.7 

14 

9.57 793 

31 

9.61 148 

36 

10.38 852 

9.96 645 

5 

5 

46 

30 

40 

18.5 

24.7 

18.0 

24.0 

17.5 

23.3 

15 

9.57 824 


9.61 184 


10.38 816 

9.96 640 


45 

50 

30.8 

30.0 

29.2 

16 

9.57 855 

31 

9.61 220 

36 

10.38 780 

9.96 634 

6 

44 







17 

9.57 885 

30 

9.61 256 

36 

10.38 744 

9.96 629 

5 

43 







18 

9.57 916 

31 

9.61 292 

36 

10.38 708 

9.96 624 

5 

42 







19 

9.57 947 

31 

9.61 328 

36 

10.38 672 

9.96 619 

5 

41 











36 











20 

9.57 978 


9.61 364 

10.38 636 

9.96 614 


40 







21 

9.58 008 

30 

9.61 400 

36 

10.38 600 

9.96 608 

6 

39 







22 

9.58 039 

31 

9.61 436 

36 

10.38 564 

9.96 603 

5 

38 







23 

9.58 070 

31 

9.61 472 

36 

10.38 528 

9.96 598 

5 

37 







24 

9.58 101 

31 

on 

9.61 508 

36 

9ft 

10.38 492 

9.96 593 

5 

5 

36 







25 

9.58 131 

OU 

9.61 544 

OO 

10.38 456 

9.96 588 


35 







26 

9.58 162 

31 

9.61 579 

35 

10.38 421 

9.96 582 

6 

34 

" 


32 


31 

30 

27 

9.58 192 

30 

9.61 615 

36 

10.38 385 

9.96 577 

5 

33 







28 

9.58 223 

31 

9.61 651 

36 

10.38 349 

9.96 572 

5 

32 

6 

n 


3.2 


3.1 

3.0 

29 

9.58 253 

30 

o 1 

9.61 687 

36 

Q CC 

10.38 313 

9.96 567 

5 

5 

31 

! 

8 


4^3 


4.1 

0.0 

4.0 

30 

9.58 284 

OI 

9.61 722 

aa 

10.38 278 

9.96 562 


30 

9 


4.8 


4.6 

4.5 

31 

9.58 314 

30 

9.61 758 

36 

10.38 242 

9.96 556 

6 

29 

10 

5.3 

1 A 7 

5.2 

5.0 

inn 

32 

9.58 345 

31 

9.61 794 

36 

10.38 206 

9.96 551 

5 

28 

20 

30 

JLU. / 
16.0 

1U.0 

15.5 

1U.U 

15.0 

33 

9.58 375 

30 

9.61 830 

36 

10.38 170 

9.96 546 

5 

27 

40 

21.3 

20.7 

20.0 

34 

9.58 406 

31 

on 

9.61 865 

35 

10.38 135 

9.96 541 

5 

5 

26 

50 

26.7 

25.8 

25.0 

35 

9.58 436 

au 

9.61 901 

ao 

10.38 099 

9.96 535 


25 







36 

9.58 467 

31 

9.61 936 

35 

10.38 064 

9.96 530 

5 

24 







37 

9.58 497 

30 

9.61 972 

36 

10.38 028 

9.96 525 

5 

23 







38 

9.58 527 

30 

9.62 008 

36 

10.37 992 

9.96 520 

5 

22 







39 

9.58 557 

30 

9.62 043 

35 

10.37 957 

9.96 514 

6 

5 

21 







40 

9.58 588 

31 

9.62 079 

ao 

10.37 921 

9.96 509 


20 







41 

9.58 618 

30 

9.62 114 

35 

10.37 886 

9.96 504 

5 

19 







42 

9.58 648 

30 

9.62 150 

36 

10.37 850 

9.96 498 

6 

18 







43 

9.58 678 

30 

9.62 185 

35 

10.37 815 

9.96 493 

5 

17 







44 

9.58 709 

31 

9.62 221 

36 

10.37 779 

9.96 488 

5 

5 

16 



29 




45 

9.58 739 

ou 

9.62 256 

ao 

10.37 744 

9.96 483 


15 

" 



6 

5 

46 

9.58 769 

30 

9.62 292 

36 

10.37 708 

9.96 477 

6 

14 

I 


9 Q 

0 6 

0.5 

47 

9.58 799 

30 

9.62 327 

35 

10.37 673 

9.96 472 

5 

13 


7 

3*' 

1 

0.7 

0.6 

48 

9.58 829 

30 

9.62 362 

35 

10.37 638 

9.96 467 

5 

12 

8 

3.9 

0.8 

0.7 

49 

9.58 859 

30 

9.62 398 

36 

35 

10.37 602 

9.96 461 

6 

5 

11 

| 

1 l 

> 

4 

4.4 
4 8 

0.9 
1 0 

0.8 

0 8 

50 

9.58 889 

30 

9.62 433 


10.37 567 

9.96 456 


10 

20 

9 '.7 

2.0 

L7 

51 

9.58 919 

30 

9.62 468 

35 

10.37 532 

9.96 451 

5 

9 

30 

14.5 

3.0 

2.5 

52 

9.58 949 

30 

9.62 504 

36 

10.37 496 

9.96 445 

6 

8 

40 

19.3 

4.0 

3.3 

53 

9.58 979 

30 

9.62 539 

35 

10.37 461 

9.96 440 

5 

7 

50 

24.2 

5.0 

4.2 

54 

9.59 009 

30 

9.62 574 

35 

10.37 426 

9.96 435 

5 

6 

6 







55 

9.59 039 

30 

9.62 609 

35 

10.37 391 

9.96 429 


5 







56 

9.59 069 

30 

9.62 645 

36 

10.37 355 

9.96 424 

5 

4 







57 

9.59 098 

29 

9.62 680 

35 

10.37 320 

9.96 419 

5 

3 







58 

9.59 128 

30 

9.62 715 

35 

10.37 285 

9.96 413 

6 

2 







59 

9.59 158 

30 

9.62 750 

35 

Q Cl 

10.37 250 

9.96 408 

5 

5 

1 







60 

9.59 188 

30 

9.62 785 

oO 

10.37 215 

9.96 403 


0 








L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


67 














































































50 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


23 ° 


/ 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

9.59 188 

30 

9.62 785 

35 

10.37 215 

9.96 403 


60 







1 

9.59 218 

9.62 820 

10.37 180 

9.96 397 

6 

59 







2 

9.59 247 

29 

9.62 855 

35 

10.37 145 

9.96 392 

5 

58 







3 

9.59 277 

30 

9.62 890 

35 

10.37 110 

9.96 387 

5 

57 







4 

9.59 307" 

30 

9.62 926 

36 

10.37 074 

9.96 381 

6 

56 









29 


35 











5 

9.59 336 

9.62 961 

10.37 039 

9.96 376 


55 







6 

9.59 366 

30 

9.62 996 

35 

10.37 004 

9.96 370 

6 

54 







7 

9.59 396 

30 

9.63 031 

35 

10.36 969 

9.96 365 

5 

53 

// 

3( 


35 

34 

8 

9.59 425 

29 

9.63 066 

35 

10.36 934 

9.96 360 

5 

52 






9 

9.59 455 

30 

9.63 101 

35 

10.36 899 

9.96 354 

6 

51 

6 

3.6 

3.5 

3.4 


29 

34 



5 







4.0 

4.5 

10 

9.59 484 

9.63 135 

10.36 865 

9.96 349 

50 

7 

8 

4.8 

*±.i 

4.7 

11 

9.59 514 

30 

9.63 170 

35 

10.36 830 

9.96 343 

6 

49 

9 

5.4 

5.2 

5.1 

12 

9.59 543 

29 

9.63 205 

35 

10.36 795 

9.96 338 

5 

48 

10 

6.0 

5.8 

5.7 

13 

9.59 573 

30 

9.63 240 

35 

10.36 760 

9.96 333 

5 

47 

20 

12.0 

11.7 

11.3 

14 

9.59 602 

29 

30 

.9.63 275 

35 

35 

10.36 725 

9.96 327 

6 

5 

46 

30 

40 

18.0 

24.0 

17.5 

23.3 

17.0 

22.7 

15 

9.59 632 

9.63 310 

10.36 690 

9.96 322 

45 

50 

30.0 

29.2 

28.3 

16 

9.59 661 

29 

9.63 345 

35 

10.36 655 

9.96 316 

6 

44 







17 

9.59 690 

29 

9.63 379 

34 

10.36 621 

9.96 311 

5 

43 







18 

9.59 720 

30 

9.63 414 

35 

10.36 586 

9.96 305 

6 

42 







19 

9.59 749 

29 

29 

9.63 449 

35 

35 

10.36 551 

9.96 300 

5 

6 

41 







20 

9.59 778 

9.63 484 

10.36 516 

9.96 294 

40 







21 

9.59 80-8 

30 

9.63 519 

35 

10.36 481 

9.96 289 

5 

39 







22 

9.59 837 

29 

9.63 553 

34 

10.36 447 

9.96 284 

5 

38 







23 

9.59 866 

29 

9.63 588 

35 

10.36 412 

9.96 278 

6 

37 







24 

9.59 895 

29 

29 

9.63 623 

35 

34 

10.36 377 

9.96 273 

5 

6 

36 







25 

9.59 924 

9.63 657 

10.36 343 

9.96 267 

35 







26 

9.59 954 

30 

9.63 692 

35 

10.36 308 

9.96 262 

5 

34 

>> 

3C 

1 

29 

28 

27 

9.59 983 

29 

9.63 726 

34 

10.36 274 

9.96 256 

6 

33 






2.8 

28 

9.60 012 

29 

9.63 761 

35 

10.36 239 

9.96 251 

5 

32 

6 

3.0 

2.9 

29 

9.60 041 

29 

29 

9.63 796 

35 

34 

10.36 204 

9.96 245 

6 

5 

31 

7 

8 

3.5 

4.0 

3.4 

3.9 

3.3 

3.7 

30 

9.60 070 

9.63 830 

10.36 170 

9.96 240 

30 

9 

4.5 

4.4 

4.2 

31 

9.60 099 

29 

9.63 865 

35 

10.36 135 

9.96 234 

6 

29 

10 

5.0 

4.8 

4.7 

32 

33 

9.60 128 
9.60 157 

29 

29 

9.63 899 
9.63 934 

34 

35 

10.36 101 
10.36 066 

9.96 229 
9.96 223 

5 

6 

28 

27 

20 

30 

40 

10.0 
15.0 
20 0 

9.7 
14.5 
19 3 

9.3 

14.0 

18 7 

34 

9.60 186 

29. 

29 

9.63 968 

34 

35 

10.36 032 

9.96 218 

5 

6 

26 

50 

25.0 

24.2 

23.3 

35 

9.60 215 

9.64 003 

10.35 997 

9.96 212 

25 







36 

9.60 244 

29 

9.64 037 

34 

10.35 963 

9.96 207 

5 

24 







37 

9.60 273 

29 

9.64 072 

35 

10.35 928 

9.96 201 

6 

23 







38 

9.60 302 

29 

9.64 106 

34 

10.35 894 

9.96 196 

5 

22 







39 

9.60 331 

29 

28 

9.64 140 

34 

35 

10.35 860 

9.96 190 

6 

5 

21 







40 

9.60 359 

9.64 175 

10.35 825 

9.96 185 

20 







41 

9.60 388 

29 

9.64 209 

34 

10.35 791 

9.96 179 

6 

19 







42 

9.60 417 

29 

9.64 243 

34 

10.35 757 

9.96 174 

5 

18 







43 

9.60 446 

29 

9.64 278 

35 

10.35 722 

9.96 168 

6 

17 







44 

9.60 474 

28 

29 

9.64 312 

34 

34 

10.35 688 

9.96 162 

6 

5 

16 







45 

9.60 503 

9.64 346 

10.35 654 

9.96 157 

15 


" 

( 


5 


46 

9.60 532 

29 

9.64 381 

35 

10.35 619 

9.96 151 

6 

14 


6 

7 





47 

9.60 561 

29 

9.64 415 

34 

10.35 585 

9.96 146 

5 

13 


u.o 

0.7 

0.8 

u.o . 

0.6 

0.7 

48 

9.60 589 

28 

9.64 449 

34 

10.35 551 

9.96 140 

6 

12 


4 

8 

49 

9.60 618 

29 

28 

9.64 483 

34 

34 

10.35 517 

9.96 135 

5 

6 

11 


9 

0.9 

0.8 

50 

9.60 646 

9.64 517 

10.35 483 

9.96 129 

10 


10 

20 

30 

1.0 

2.0 

3.0 

0.8 

1.7 

2.5 

51 

9.60 675 

29 

9.64 552 

35 

10.35 448 

9.96 123 

6 

9 


52 

9.60 704 

29 

9.64 586 

34 

10.35 414 

9.96 118 

5 

8 


40 

4.0 

3.3 

53 

9.60 732 

28 

9.64 620 

34 

10.35 380 

9.96 112 

6 

7 


50 

5.0 

4.2 

54 

9.60 761 

29 

28 

9.64 654 

34 

34 

10.35 346 

9.96 107 

5 

6 

6 







55 

9.60 789 

9.64 688 

10.35 312 

9.96 101 

5 







56 

9.60 818 

29 

9.64 722 

34 

10.35 278 

9.96 095 

6 

4 







57 

9.60 846 

28 

9.64 756 

34 

10.35 244 

9.96 090 

5 

3 







58 

9.60 875 

29 

9.64 790 

34 

10.35 210 

9.96 084 

6 

2 







59 

9.60 903 

28 

28 

9.64 824 

34 

Q4 

10.35 176 

9.96 079 

5 

6 

1 







60 

9.60 931 

9.64 858 


10.35 142 

9.96 073 

0 








L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

f 

Prop. Pts. 


66 ' 













































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


51 


24 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.60 931 
9.60 960 

9.60 988 

9.61 016 
9.61 045 

29 

28 

28 

29 

28 

28 

28 

29 

28 

28 

28 

28 

28 

28 

28 

28 

29 

27 

28 
28 

28 

28 

28 

28 

28 

28 

27 

28 
28 
28 

27 

28 
28 

27 

28 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 
27 

27 

28 
27 

27 

27 

28 
27 
27 

27 

27 

28 
27 
27 

9.64 858 
9.64 892 
9.64 926 
9.64 960 
9.64 994 

34 

34 

34 

34 

34 

34 

34 

34 

34 

33 

34 
34 
34 
34 

33 

34 
34 

33 

34 
34 

33 

34 
34 

33 

34 

33 

34 

33 

34 

33 

34 

33 

34 

33 

34 

33 

33 

34 
33 

33 

34 
33 
33 

33 

34 

33 

33 

33 

33 

34 

33 

33 

33 

33 

33 

33 

33 

33 

33 

33 

10.35 142 
10.35 108 
10.35 074 
10.35 040 
10.35 006 

9.96 073 
9.96 067 
9.96 062 
9.96 056 
9.96 050 

6 

5 

6 

6 

5 

6 

5 

6 

6 

5 

6 

6 

5 

6 

6 

6 

5 

6 

6 

5 

6 

6 

6 

5 

6 

6 

5 

6 

6 

6 

5 

6 

6 

6 

6 

5 

6 

6 

6 

6 

5 

6 

6 

6 

6 

5 

6 

6 

6 

6 

6 

5 

6 

6 

6 

6 

6 

6 

6 

5 

60 

59 

58 

57 

56 

" 34 33 

6 3.4 3.3 

7 4.0 3.8 

8 4.5 4.4 

9 5.1 5.0 

10 5.7 5.5 

20 11.3 11.0 

30 17.0 16.5 

40 22.7 22.0 

50 28.3 27.5 

" 29 28 27 

6 2.9 2.8 2.7 

7 3.4 3.3 3.2 

8 3.9 3.7 3.6 

9 4.4 4.2 4.0 

10 4.8 4.7 4.5 

20 9.7 9.3 9.0 

30 14.5 14.0 13.5 

40 19.3 18.7 18.0 

50 24.2 23.3 22.5 

"65 

6 0.6 0.5 

7 0.7 0.6 

8 0.8 0.7 

9 0.9 0.8 

10 1.0 0.8 

20 2.0 1.7 

30 3.0 2.5 

40 4.0 3.3 

50 5.0 4.2 

5 

6 

7 

8 

9 

9.61 073 
9.61 101 
9.61 129 
9.61 158 
9.61 186 

9.65 028 
9.65 062 
9.65 096 
9.65 130 
9.65 164 

10.34 972 
10.34 938 
10.34 904 
10.34 870 
10.34 836 

9.96 045 
9.96 039 
9.96 034 
9.96 028 
9.96 022 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.61 214 
9.61 242 
9.61 270 
9.61 298 
9.61 326 

9.65 197 
9.65 231 
9.65 265 
9.65 299 
9.65 333 

10.34 803 
10.34 769 
10.34 735 
10.34 701 
10.34 667 

9.96 017 
9.96 011 
9.96 005 
9.96 000 
9.95 994 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.61 354 
9.61 382 
9.61411 
9.61 438 
9.61 466 

9.65 366 
9.65 400 
9.65 434 
9.65 467 
9.65 501 

10.34 634 
10.34 600 
10.34 566 
10.34 533 
10.34 499 

9.95 988 
9.95 982 
9.95 977 
9.95 971 
9.95 965 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.61 494 
9.61 522 
9.61 550 
9.61 578 
9.61 606 

9.65 535 
9.65 568 
9.65 602 
9.65 636 
9.65 669 

10.34 465 
10.34 432 
10.34 398 
10.34 364 
10.34 331 

9.95 960 
9.95 954 
9.95 948 
9.95 942 
9.95 937 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.61 634 
9.61 662 
9.61 689 
9.61 717 
9.61 745 

9.65 703 
9.65 736 
9.65 770 
9.65 803 
9.65 837 

10.34 297 
10.34 264 
10.34 230 
10.34 197 
10.34 163 

9.95 931 
9.95 925 
9.95 920 
9.95 914 
9.95 908 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.61 773 
9.61 800 
9.61 828 
9.61 856 
9.61 883 

9.65 870 
9.65 904 
9.65 937 

9.65 971 

9.66 004 

10.34 130 
10.34 096 
10.34 063 
10.34 029 
10.33 996 

9.95 902 
9.95 897 
9.95 891 
9.95 885 
9.95 879 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.61 911 
9.61 939 
9.61 966 

9.61 994 

9.62 021 

9.66 038 
9.66 071 
9.66 104 
9.66 138 
9.66 171 

10.33 962 
10.33 929 
10.33 896 
10.33 862 
10.33 829 

9.95 873 
9.95 868 
9.95 862 
9.95 856 
9.95 850 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 
45' 

46 

47 

48 

49 

9.62 049 
9.62 076 
9.62 104 
9.62 131 
9.62 159 

9.66 204 
9.66 238 
9.66 271 
9.66 304 
9.66 337 

10.33 796 
10.33 762 
10.33 729 
10.33 696 
10.33 663 

9.95 844 
9.95 839 
9.95 833 
9.95 827 
9.95 821 

20 

19 

18 

17 

16 

9.62 186 
9.62 214 
9.62 241 
9.62 268 
9.62 296 

9.66 371 
9.66 404 
9.66 437 
9.66 470 
9.66 503 

10.33 629 
10.33 596 
10.33 563 
10.33 530 
10.33 497 

9.95 815 
9.95 810 
9.95 804 
9.95 798 
9.95 792 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.62 323 
9.62 350 
9.62 377 
9.62 405 
9.62 432 

9.66 537 
9.66 570 
9.66 603 
9.66 636 
9.66 669 

10.33 463 
10.33 430 
10.33 397 
10.33 364 
10.33 331 

9.95 786 
9.95 780 
9.95 775 
9.95 769 
9.95 763 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.62 459 
9.62 486 
9.62 513 
9.62 541 
9.62 568 

9.66 702 
9.66 735 
9.66 768 
9.66 801 
9.66 834 

10.33 298 
10.33 265 
10.33 232 
10.33 199 
10.33 166 

9.95 757 
9.95 751 
9.95 745 
9.95 739 
9.95 733 

5 

4 

3 

2 

1 

60 

9.62 595 

9.66 867 

10.33 133 

9.95 728 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


65 ‘ 









































































52 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


25 ° 


' 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.62 595 
9.62 622 
9.62 649 
9.62 676 
9.62 703 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

26 

27 

27 

27 

27 

26 

27 

27 

27 

26 

27 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 

27. 

26 

26 

27 

26 

26 

27 

26 

26 

26 

27 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

9.66 867 
9.66 900 
9.66 933 
9.66 966 
9.66 999 

33 

33 

33 

33 

33 

33 

33 

33 

32 

33 

33 

33 

33 

32 

33 

33 

33 

32 

33 
33 

32 

33 
33 

32 

33 

32 

33 
33 

32 

33 

32 

33 

32 

33 
32 

32 

33 

32 

33 
32 

32 

33 
32 

32 

33 

32 

32 

32 

33 
32 

32 

32 

32 

33 
32 

32 

32 

32 

32 

32 

10.33 133 
10.33 100 
10.33 067 
10.33 034 
10.33 001 

9.95 728 
9.95 722 
9.95 716 
9.95 710 
9.95 704 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

5 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

7 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

7 

6 

« 

6 

6 

6 

6 

7 

6 

6 

6 

60 

59 

58 

57 

56 

" 33 32 

6 3.3 3.2 

7 3.8 3.7 

8 4.4 4.3 

8 5.0 4.8 

10 5.5 5.3 

20 11.0 10.7 

30 16.5 16.0 

40 22.0 21.3 

50 27.5 26.7 

" 27 26 

6 2.7 2.6 

7 3.2 3.0 

8 3.6 3.5 

9 4.0 3.9 

10 4.5 4.3 

20 9.0 8.7 

30 13.5 13.0 

40 18.0 17.3 

50 22.5 21.7 

"765 

6 0.7 0.6 0.5 

7 0.8 0.7 0.6 

8 0.9 0.8 0.7 

9 1.0 0.9 0.8 

10 1.2 1.0 0.8 

20 2.3 2.0 1.7 

30 3.5 3.0 2.5 

40 4.7 4.0 3.3 

50 5.8 5.0 4.2 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 
19 

9.62 730 
9.62 757 
9.62 784 
9.62 811 
9.62 838 

9.67 032 
9.67 065 
9.67 098 
9.67 131 
9.67 163 

10.32 968 
10.32 935 
10.32 902 
10.32 869 
10.32 837 

9.95 698 

9.95 692 
9.95 686 
9.95 680 
9.95 674 

55 

54 

53 

52 

51 

9.62 865 
9.62 892 
9.62 918 
9.62 945 
9.62 972 

9.67 196 
9.67 229 
9.67 262 
9.67 295 
9.67 327 

10.32 804 
10.32 771 
10.32 738 
10.32 705 
10.32 673 

9.95 668 
9.95 663 
9.95 657 
9.95 651 
9.95 645 

50 

49 

48 

47 

46 

9.62 999 

9.63 026 
9.63 052 
9.63 079 
9.63 106 

9.67 360 
9.67 393 
9.67 426 
9.67 458 
9.67 491 

10.32 640 
10.32 607 
10.32 574 
10.32 542 
10.32 509 

9.95 639 
9.95 633 
9.95 627 
9.95 621 
9.95 615 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.63 133 
9.63 159 
9.63 186 
9.63 213 
9.63 239 

9.67 524 
9.67 556 
9.67 589 
9.67 622 
9.67 654 

10.32 476 
10.32 444 
10.32 411 
10.32 378 
10.32 346 

9.95 609 
9.95 603 
9.95 597 
9.95 591 
9.95 585 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.63 266 
9.63 292 
9.63 319 
9.63 345 
9.63 372 

9.67 687 
9.67 719 
9.67 752 
9.67 785 
9.67 817 

10.32 313 
10.32 281 
10.32 248 
10.32 215 
10.32 183 

9.95 579 
9.95 573 
9.95 567 
9.95 561 
9.95 555 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.63 398 
9.63 425 
9.63 451 
9.63 478 
9.63 504 

9.67 850 
9.67 882 
9.67 915 
9.67 947 
9.67 980 

10.32 150 
10.32 118 
10.32 085 
10.32 053 
10.32 020 

9.95 549 
9.95 543 
9.95 537 
9.95 531 
9.95 525 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.63 531 
9.63 557 
9.63 583 
9.63 610 
9.63 636 

9.68 012 
9.68 044 
9.68 077 
9.68 109 
9.68 142 

10.31 988 
10.31 956 
10.31 923 
10.31 891 
10.31 858 

9.95 519 
9.95 513 
9.95 507 
9.95 500 
9.95 494 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.63 662 
9.63 689 
9.63 715 
9.63 741 
9.63 767 

9.68 174 
9.68 206 
9.68 239 
9.68 271 
9.68 303 

10.31 826 
10.31 794 
10.31 761 
10.31 729 
10.31 697 

9.95 488 
9.95 482 
9.95 476 
9.95 470 
9.95 464 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.63 794 
9.63 820 
9.63 846 
9.63 872 
9.63 898 

9.68 336 
9.68 368 
9.68 400 
9.68 432 
9.68 465 

10.31 664 
10.31 632 
10.31 600 
10.31 568 
10.31 535 

9.95 458 
9.95 452 
9.95 446 
9.95 440 
9.95 434 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.63 924 
9.63 950 

9.63 976 

9.64 002 
9.64 028 

9.68 497 
9.68 529 
9.68 561 
9.68 593 
9.68 626 

10.31 503 
10.31 471 
10.31 439 
10.31 407 
10.31 374 

9.95 427 
9.95 421 
9.95 415 
9.95 409 
9.95 403 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.64 054 
9.64 080 
9.64 106 
9.64 132 
9.64 158 

9.68 658 
9.68 690 
9.68 722 
9.68 754 
9.68 786 

10.31 342 
10.31 310 
10.31 278 
10.31 246 
10.31 214 

9.95 397 
9.95 391 
9.95 384 
9.95 378 
9.95 372 

5 

4 

3 

2 

1 

60 

9.64 184 

9.68 818 

10.31 182 

9.95 366 

0 


L Cos 

d 

L Cot 

c d 

| L Tan 

L Sin 

d 

' 

Prop. Pts. 


81 ' 









































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


53 


26 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.64 184 
9.64 210 
9.64 236 
9.64 262 
9.64 288 

26 

26 

26 

26 

25 

26 
26 
26 
26 

25 

26 
26 

25 

26 
26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 
25 

25 

26 

25 

25 

26 
25 
25 

25 

26 
25 
25 
25 

25 

25 

26 
25 
25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

24 

25 
25 
25 
25 
25 

9.68 818 
9.68 850 
9.68 882 
9.68 914 
9.68 946 

9.68 978 

9.69 010 
9.69 042 
9.69 074 
9.69 106 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

32 

31 

32 
32 
32 
32 

31 

32 
32 
32 

31 

32 
32 

31 

32 
32 

31 

32 

31 

32 
32 

31 

32 

31 

32 

31 

32 

31 

32 

31 

32 

31 

31 

32 

31 

32 

31 

31 

32 
31 

31 

32 
31 
31 

31 

32 

10.31 182 
10.31 150 
10.31 118 
10.31 086 
10.31 054 

9.95 366 
9.95 360 
9.95 354 
9.95 348 
9.95 341 

6 

6 

6 

7 

6 

6 

6 

6 

7 

6 

6 

6 

6 

7 

6 

6 

6 

7 

6 

6 

6 

7 

6 

6 

6 

7 

6 

6 

7 

6 

6 

6 

7 

6 

6 

7 

6 

6 

7 

6 

6 

7 

6 

7 

6 

6 

7 

6 

6 

7 

6 

7 

6 

6 

7 

6 

7 

6 

6 

7 

60 

59 

58 

57 

56 

" 32 31 

6 3.2 3.1 

7 3.7 3.6 

8 4.3 4.1 

9 4.8 4.6 

10 5.3 5.2 

20 10.7 10.3 

30 16.0 15.5 

40 21.3 20.7 

50 26.7 25.8 

" 26 25 24 

6 2.6 2.5 2.4 

7 3.0 2.9 2.8 

8 3.5 3.3 3.2 

9 3.9 3.8 3.6 

10 4.3 4.2 4.0 

20 8.7 8.3 8.0 

30 13.0 12.5 12.0 

40 17.3 16.7 16.0 

50 21.7 20.8 20.0 

"76 

6 0.7 0.6 

7 0.8 0.7 

8 0.9 0.8 

9 1.0 0.9 

10 1.2 1.0 

20 2.3 2.0 

30 3.5 3.0 

40 4.7 4.0 

50 5.8 5.0 

5 

6 

7 

8 

9 

9.64 313 
9.64 339 
9.64 365 
9.64 391 
9.64 417 

10.31 022 
10.30 990 
10.30 958 
10.30 926 
10.30 894 

9.95 335 
9.95 329 
9.95 323 
9.95 317 
9.95 310 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.64 442 
9.64 468 
9.64 494 
9.64 519 
9.64 545 

9.69 138 
9.69 170 
9.69 202 
9.69 234 
9.68 266 

10.30 862 
10.30 830 
10.30 798 
10.30 766 
10.30 734 

9.95 304 
9.95 298 
9.95 292 
9.95 286 
9.95 279 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.64 571 
9.64 596 
9.64 622 
9.64 647 
9.64 673 

9.69 298 
9.69 329 
9.69 361 
9.69 393 
9.69 425 

10.30 702 
10.30 671 
10.30 639 
10.30 607 
10.30 575 

9.95 273 
9.95 267 
9.95 261 
9.95 254 
9.95 248 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.64 698 
9.64 724 
9.64 749 
9.64 775 
9.64 800 

9.69 457 
9.69 488 
9.69 520 
9.69 552 
9.69 584 

10.30 543 
10.30 512 
10.30 480 
10.30 448 
10.30 416 

9.95 242 
9.95 236 
9.95 229 
9.95 223 
9.95 217 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.64 826 
9.64 851 
9.64 877 
9.64 902 
9.64 927 

9.69 615 
9.69 647 
9.69 679 
9.69 710 
9.69 742 

10.30 385 
10.30 353 
10.30 321 
10.30 290 
10.30 258 

9.95 211 
9.95 204 
9.95 198 
9.95 192 
9.95 185 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.64 953 

9.64 978 

9.65 003 
9.65 029 
9.65 054 

9.69 774 
9.69 805 
9.69 837 
9.69 868 
9.69 900 

10.30 226 
10.30 195 
10.30 163 
10.30 132 
10.30 100 

9.95 179 
9.95 173 
9.95 167 
9.95 160 
9.95 154 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.65 079 
9.65 104 
9.65 130 
9.65 155 
9.65 180 

9.69 932 
9.69 963 

9.69 995 

9.70 026 
9.70 058 

10.30 068 
10.30 037 
10.30 005 
10.29 974 
10.29 942 

9.95 148 
9.95 141 
9.95 135 
9.95 129 
9.95 122 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.65 205 
9.65 230 
9.65 255 
9.65 281 
9.65 306 

9.70 089 
9.70 121 
9.70 152 
9.70 184 
9.70 215 

10.29 911 
10.29 879 
10.29 848 
10.29 816 
10.29 785 

9.95 116 
9.95 110 
9.95 103 
9.95 097 
9.95 090 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.65 331 
9.65 356 
9.65 381 
9.65 406 
9.65 431 

9.70 247 
9.70 278 
9.70 309 
9.70 341 
9.70 372 

10.29 753 
10.29 722 
10.29 691 
10.29 659 
10.29 628 

9.95 084 
9.95 078 
9.95 071 
9.95 065 
9.95 059 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.65 456 
9.65 481 
9.65 506 
9.65 531 
9.65 556 

9.70 404 
9.70 435 
9.70 466 
9.70 498 
9.70 529 

10.29 596 
10.29 565 
10.29 534 
10.29 502 
10.29 471 

9.95 052 

9.95 046 
9.95 039 
9.95 033 
9.95 027 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.65 580 
9.65 605 
9.65 630 
9.65 655 
9.65 680 

9.70 560 
9.70 592 
9.70 623 
9.70 654 
9.70 685 

10.29 440 
10.29 408 
10.29 377 
10.29 346 
10.29 315 

9.95 020 

9.95 014 
9.95 007 
9.95 001 
9.94 995 

5 

4 

3 

2 

1 

60 

9.65 705 

9.70 717 

10.29 283 

9.94 988 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


63 ’ 












































































54 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


27 ° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.65 705 

24 

9.70 717 

31 

10.29 283 

9.94 

988 


60 

1 

9.65 729 

9.70 748 

10.29 252 

9.94 

982 

6 

59 

2 

9.65 754 

25 

9.70 779 

31 

10.29 221 

9.94 

975 

7 

58 

3 

9.65 779 

25 

9.70 810 

31 

10.29 190 

9.94 

969 

6 

57 

4 

9.65 804 

25 

9.70 841 

31 

•lO 

10.29 159 

9.94 

962 

7 

6 

56 

5 

9.65 828 


9.70 873 

oz 

31 

10.29 127 

9.94 

956 


55 

6 

9.65 853 

25 

9.70 904 

10.29 096 

9.94 

949 

7 

45 

7 

9.65 878 

25 

9.70 935 

31 

10.29 065 

9.94 

943 

6 

53 

8 

9.65 902 

24 

9.70 966 

31 

10.29 034 

9.94 

936 

7 

52 

9 

9.65 927 

25 

9.70 997 

31 

qi 

10.29 003 

9.94 

930 

6 

7 

51 

10 

9.65 952 


9.71 028 

o 1 

10.28 972 

9.94 

923 


50 

11 

9.65 976 

24 

9.71 059 

31 

10.28 941 

9.94 

917 

6 

49 

12 

9.66 001 

25 

9.71 090 

31 

10.28 910 

9.94 

911 

6 

48 

13 

9.66 025 

24 

9.71 121 

31 

10.28 879 

9.94 

904 

7. 

47 

14 

9.66 050 

25 

9.71 153 

32 

10.28 847 

9.94 

898 

6 

7 

46 

15 

9.66 075 


9.71 184 

O X 

31 

10.28 816 

9.94 

891 

6 

45 

16 

9.66 099 

24 

9.71 215 

10.28 785 

9.94 

885 

44 

17 

9.66 124 

25 

9.71 246 

31 

10.28 754 

9.94 

878 

7 

43 

18 

9.66 148 

24 

9.71 277 

31 

10.28 723 

9.94 

871 

7 

42 

19 

9.66 173 

25 

9.71 308 

31 

91 

10.28 692 

9.94 

865 

6 

7 

41 

20 

9.66 197 


9.71 339 

OX 

10.28 661 

9.94 

858 


40 

21 

9.66 221 

24 

9.71 370 

31 

10.28 630 

9.94 

852 

6 

39 

22 

9.66 246 

25 

9.71 401 

31 

10.28 599 

9.94 

845 

7 

38 

23 

9.66 270 

24 

9.71 431 

30 

10.28 569 

9.94 

839 

6 

37 

24 

9.66 295 

25 

OA 

9.71 462 

31 

91 

10.28 538 

9.94 

832 

7 

6 

36 

25 

9.66 319 

Z^k 

9.71 493 

OX 

10.28 507 

9.94 

826 


35 

26 

9.66 343 

24 

9.71 524 

31 

10.28 476 

9.94 

819 

7 

34 

27 

9.66 368 

25 

9.71 555 

31 

10.28 445 

9.94 

813 

6 

33 

28 

9.66 392 

24 

9.71 586 

31 

10.28 414 

9.94 

806 

7 

32 

29 

9.66 416 

24 

O K 

9.71 617 

31 

31 

10.28 383 

9.94 

799 

7 

6 

31 

30 

9.66 441 

zo 

9.71 648 

10.28 352 

9.94 

793 

30 

31 

9.66 465 

24 

9.71 679 

31 

10.28 321 

9.94 

786 

7 

29 

32 

9.66 489 

24 

9.71 709 

30 

10.28 291 

9.94 

780 

6 

28 

33 

9.66 513 

24 

9.71 740 

31 

10.28 260 

9.94 

773 

7 

27 

34 

9.66 537 

24 

O C 

9.71 771 

31 

91 

10.28 229 

9.94 

767 

6 

7 

26 

35 

9.66 562 


9.71 802 

ox 

10.28 198 

9.94 

760 


25 

36 

9.66 586 

24 

9.71 833 

31 

10.28 167 

9.94 

753 

7 

24 

37 

9.66 610 

24 

9.71 863 

30 

10.28 137 

9.94 

747 

6 

23 

38 

9.66 634 

24 

9.71 894 

31 

10.28 106 

9.94 

740 

7 

22 

39 

9.66 658 

24 

OA 

9.71 925 

31 

9f» 

10.28 075 

9.94 

734 

6 

7 

21 

40 

9.66 682 

Zk 

9.71 955 

ou 

10.28 045 

9.94 

727 


20 

41 

9.66 706 

24 

9.71 986 

31 

10.28 014 

9.94 

720 

7 

19 

42 

9.66 731 

25 

9.72 017 

31 

10.27 983 

9.94 

714 

6 

18 

43 

9.66 755 

24 

9.72 048 

31 

10.27 952 

9.94 

707 

7 

17 

44 

9.66 779 

24 

OA 

9.72 078 

30 

31 

10.27 922 

9.94 

700 

7 

g 

16 

45 

9.66 803 

Zrk 

9.72 109 

10.27 891 

9.94 

694 


15 

46 

9.66 827 

24 

9.72 140 

31 

10.27 860 

9.94 

687 

7 

14 

47 

9.66 851 

24 

9.72 170 

30 

10.27 830 

9.94 

680 

7 

13 

48 

9.66 875 

24 

9.72 201 

31 

10.27 799 

9.94 

674 

6 

12 

49 

9.66 899 

24 

OQ 

9.72 231 

30 

91 

10.27 769 

9.94 

667 

7 

7 

11 

50 

9.66 922 

Zo 

9.72 262 

ox 

10.27 738 

9.94 

660 


10 

51 

9.66 946 

24 

9.72 293 

31 

10.27 707 

9.94 

654 

6 

9 

52 

9.66 970 

24 

9.72 323 

30 

10.27 677 

9.94 

647 

7 

8 

53 

9.66 994 

24 

9.72 354 

31 

10.27 646 

9.94 

640 

7 

7 

54 

9.67 018 

24 

24 

9.72 384 

30 

91 

10.27 616 

9.94 

634 

6 

7 

6 

55 

9.67 042 

9.72 415 

O X 

10.27 585 

9.94 

627 


5 

56 

9.67 066 

24 

9.72 445 

30 

10.27 555 

9.94 

620 

7 

4 

57 

9.67 090 

24 

9.72 476 

31 

10.27 524 

9.94 

614 

6 

3 

58 

9.67 113 

23 

9.72 506 

30 

10.27 494 

9.94 

607 

7 

2 

59 

9.67 137 

24 

24 

9.72 537 

31 

30 

10.27 463 

9.94 

600 

7 

7 

1 

60 

9.67 161 

9.72 567 

10.27 433 

9.94 

:593 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

f 


Prop. Pts. 


" 

32 

31 

6 

3.2 

3.1 

7 

3.7 

3.6 

8 

4.3 

4.1 

9 

4.8 

4.6 

10 

5.3 

5.2 

20 

10.7 

10.3 

30 

16.0 

15.5 

40 

21.3 

20.7 

50 

26.7 

25.8 


3.0 

3.5 
4.0 

4.5 
5.0 

10.0 

15.0 

20.0 

25.0 


" 

25 

24 

23 

6 

2.5 

2.4 

2.3 

7 

2.9 

2.8 

2.7 

8 

3.3 

3.2 

3.1 

9 

3.8 

3.6 

3.4 

10 

4.2 

4.0 

3.8 

20 

8.3 

8.0 

7.7 

30 

12.5 

12.0 

11.5 

40 

16.7 

16.0 

15.3 

50 

20.8 

20.0 

19.2 


Prop. Pts. 


62 













































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


55 


28 ° 


7 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.67 161 


9.72 567 


10.27 433 

9.94 593 


60 

1 

9.67 185 

24 

9.72 598 

31 

10.27 402 

9.94 587 

6 

59 

2 

9.67 208 

23 

9.72 628 

30 

10.27 372 

9.94 580 

7 

58 

3 

9.67 232 

24 

9.72 659 

31 

10.27 341 

9.94 573 

7 

57 

4 

9.67 256 

24 

24 

9.72 689 

30 

31 

10.27 311 

9.94 567 

6 

•7 

56 

5 

9.67 280 

9.72 720 

10.27 280 

9.94 560 

l 

55 

6 

9.67 303 

23 

9.72 750 

30 

10.27 250 

9.94 553 

7 

54 

7 

9.67 327 

24 

9.72 780 

30 

10.27 220 

9.94 546 

7 

53 

8 

9.67 350 

23 

9.72 811 

31 

10.27 189 

9.94 540 

6 

52 

9 

9.67 374 

24 

24 

9.72 841 

30 

31 

10.27 159 

9.94 533 

7 

n 

51 

10 

9.67 398 

9.72 872 

10.27 128 

9.94 526 

7 

50 

11 

9.67 421 

23 

9.72 902 

30 

10.27 098 

9.94 519 

7 

49 

12 

9.67 445 

24 

9.72 932 

30 

10.27 068 

9.94 513 

6 

48 

13 

9.67 468 

23 

9.72 963 

31 

10.27 037 

9.94 506 

7 

47 

14 

9.67 492 

24 

23 

9.72 993 

30 

30 

10.27 007 

9.94 499 

7 

7 

46 

15 

9.67 515 

9.73 023 

10.26 977 

9.94 492 

i 

45 

16 

9.67 539 

24 

9.73 054 

31 

10.26 946 

9.94 485 

7 

44 

17 

9.67 562 

23 

9.73 084 

30 

10.26 916 

9.94 479 

6 

43 

18 

9.67 586 

24 

9.73 114 

30 

10.26 886’ 

9.94 472 

7 

42 

19 

9.67 609 

23 

24 

0.73 144 

30 

31 

10.26 856 

9.94 465 

7 

7 

41 

20 

9.67 633 

9.73 175 

10.26 825 

9.94 458 

/ 

40 

21 

9.67 656 

23 

9.73 205 

30 

10.26 795 

9.94 451 

7 

39 

22 

9.67 680 

24 

9.73 235 

30 

10.26 765 

9.94 445 

6 

38 

23 

9.67 703 

23 

9.73 265 

30 

10.26 735 

9.94 438 

7 

37 

24 

9.67 726 

23 

24 

9.73 295 

30 

31 

10.26 705 

9.94 431 

7 

7 

36 

25 

9.67 750 

9.73 326 

10.26 674 

9.94 424 

t 

35 

26 

9.67 773 

23 

9.73 356 

30 

10.26 644 

9.94 417 

7 

34 

27 

9.67 796 

23' 

9.73 386 

30 

10.26 614 

9.94 410 

7 

33 

28 

9.67 820 

24 

9.73 416 

30 

10.26 584 

9.94 404 

6 

32 

29 

9.67 843 

23 

23 

9.73 446 

30 

30 

10.26 554 

9.94 397 

7 

7 

31 

30 

9.67 866 

9.73 476 

10.26 524 

9.94 390 


30 

31 

9.67 890 

24 

9.73 507 

31 

10.26 493 

9.94 383 

7 

29 

32 

9.67 913 

23 

9.73 537 

30 

10.26 463 

9.94 376 

7 

28 

33 

9.67 936 

23 

9.73 567 

30 

10.26 433 

9.94 369 

7 

27 

34 

9.67 959 

23 

9.73 597 

30 

qa 

10.26 403 

9.94 362 

7 

7 

26 

35 

9.67 982 

— 9 

9.73 627 

OU 

10.26 373 

9.94 355 


25 

36 

9.68 006 

24 

9.73 657 

30 

10.26 343 

9.94 349 

6 

24 

37 

9.68 029 

23 

9.73 687 

30 

10.26 313 

9.94 342 

7 

23 

38 

9.68 052 

23 

9.73 717 

30 

10.26 283 

9.94 335 

7 

22 

39 

9.68 075 

23 

93 

9.73 747 

30 
q a 

10.26 253 

9.94 328 

7 

7 

21 

40 

9.68 098 


9.73 777 

OKJ 

10.26 223 

9.94 321 


20 

41 

9.68 121 

23 

9.73 807 

30 

10.26 193 

9.94 314 

7 

19 

42 

9.68 144 

23 

9.73 837 

30 

10.26 163 

9.94 307 

7 

18 

43 

9.68 167 

23 

9.73 867 

30 

10.26 133 

9.94 300 

7 

17 

44 

9.68 190 

23 

OQ 

9.73 897 

30 

10.26 103 

9.94 293 

7 

7 

16 

45 

9.68 213 


9.73 927 

OU 

10.26 073 

9.94 286 


15 

46 

9.68 237 

24 

9.73 957 

30 

10.26 043 

9.94 279 

7 

14 

47 

9.68 260 

23 

9.73 987 

30 

10.26 013 

9.94 273 

6 

13 

48 

9.68 283 

23 

9.74 017 

30 

10.25 983 

9.94 266 

7 

12 

49 

9.68 305 

22 

OQ 

9.74 047 

30 

o A 

10.25 953 

9.94 259 

7 

7 

11 

50 

9.68 328 


9.74 077 

OU 

10.25 923 

9.94 252 


10 

51 

9.68 351 

23 

9.74 107 

30 

10.25 893 

9.94 245 

7 

9 

52 

9.68 374 

23 

9.74 137 

30 

10.25 863 

9.94 238 

7 

8 

53 

9.68 397 

23 

9.74 166 

29 

10.25 834 

9.94 231 

7 

7 

54 

9.68 420 

23 

OQ 

9.74 196 

30 

Oft 

10.25 804 

9.94 224 

7 

7 

6 

55 

9.68 443 


9.74 226 

OU 

10.25 774 

9.94 217 


5 

56 

9.68 466 

23 

9.74 256 

30 

10.25 744 

9.94 210 

7 

4 

57 

9.68 489 

23 

9.74 286 

30 

10.25 714 

9.94 203 

7 

3 

58 

9.68 512 

23 

9.74 316 

30 

10.25 684 

9.94 196 

7 

2 

59 

9.68 534 

22 

9.74 345 

29 

Oft 

10.25 655 

9.94 189 

7 

7 

1 

60 

9.68 557 

23 

9.74 375 

OU 

10.25 625 

9.94 182 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

’ 


Prop. Pts. 


" 

31 

30 

29 

6 

3.1 

3.0 

2.9 

7 

3.6 

3.5 

3.4 

8 

4.1 

4.0 

3.9 

9 

4.6 

4.5 

4.4 

10 

5.2 

5.0 

4.8 

20 

10.3 

10.0 

9.7 

30 

15.5 

15.0 

14.5 

40 

20.7 

20.0 

19.3 

50 

25.8 

25.0 

24.2 


" 

24 

23 

22 

6 

2.4 

2.3 

2.2 

7 

2.8 

2.7 

2.6 

8 

3.2 

3.1 

2.9 

9 

3.6 

3.4 

3.3 

10 

4.0 

3.8 

3.7 

20 

8.0 

7.7 

7.3 

30 

12.0 

11.5 

11.0 

40 

16.0 

15.3 

14.7 

50 

20.0 

19.2 

18.3 


Prop. Pts. 


61 









































































56 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 

29 ° 


VI 


/ 

L Sin 

d 

L Teh 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.68 557 
9.68 580 
9.68 603 
9.68 625 
9.68 648 

23 

23 

22 

23 

23 

23 

22 

23 

23 

22 

23 

22 

23 

23 

22 

23 

22 

23 

22 

23 

22 

23 

22 

23 

22 

22 

23 

22 

23 

22 

22 

23 

22 

22 

22 

23 

22 

22 

22 

22 

23 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

9.74 375 
9.74 405 
9.74 435 
9.74 465 
9.74 494 

30 

30 

30 

29 

30 

30 

29 

30 
30 
30 

29 

30 
30 

29 

30 

30 

29 

30 

29 

30 

29 

30 
30 

29 

30 

29 

30 

29 

30 

29 

30 

29 

30 
29 

29 

30 

29 

30 
29 

29 

30 

29 

30 
29 

29 

30 
29 
29 

29 

30 

29 

29 

29 

30 
29 

29 

29 

30 
29 
29 

10.25 625 

10.25 595 
10.25 565 
10.25 535 
10.25 506 

9.94 182 
9.94 175 
9.94 168 
9.94 161 
9.94 154 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

8 

7 

7 

7 

7 

7 

7 

7 

8 

7 

7 

7 

7 

7 

8 

7 

7 

7 

7 

8 

7 

7 

7 

8 

7 

7 

7 

7 

8 

7 

7 

8 

7 

7 

7 

8 

7 

60 

59 

58 

57 

56 

" 30 29 

6 3.0 2.9 

7 3.5 3.5 

8 4.0 3.9 

9 4.5 4.4 

10 5.0 4.8 

20 10.0 9.7 

30 15.0 14.5 

40 20.0 19.3 

50 25.0 24.2 

" 23 22 

6 2.3 2.2 

7 2.7 2.6 

8 3.1 2.9 

9 3.4 3.3 

10 3.8 3.7 

20 7.7 7.3 

30 11.5 11.0 

40 15.3 14.7 

50 19.2 18.3 

"87 

6 0.8 0.7 

7 0.9 0.8 

8 1.1 0.9 

9 1.2 1.0 

10 1.3 1.2 

20 2.7 2.3 

30 4.0 3.5 

40 5.3 4.7 

50 6 7 5.8 

5 

6 

7 

8 

9 

9.68 671 
9.68 694 
9.68 716 
9.68 739 
9.68 762 

9.74 524 
9.74 554 
9.74 583 
9.74 613 
9.74 643 

10.25 476 
10.25 446 
10.25 417 
10.25 387 
10.25 357 

9.94 147 
9.94 140 
9.94 133 
9.94 126 
9.94 119 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

15 

16 

17 

18 
19 

9.68 784 
9.68 807 
9.68 829 
9.68 852 
9.68 875 

9.74 673 
9.74 702 
9.74 732 
9.74 762 
9.74 791 

10.25 327 
10.25 298 
10.25 268 
10.25 238 
10.25 209 

9.94 112 
9.94 105 
9.94 098 
9.94 090 
9.94 083 

50 

49 

48 

47 

46 

9.68 897 
9.68 920 
9.68 942 
9.68 965 
9.68 987 

9.74 821 
9.74 851 
9.74 880 
9.74 910 
9.74 939 

10.25 179 
10.25 149 
10.25 120 
10.25 090 
10.25 061 

9.94 076 
9.94 069 
9.94 062 
9.94 055 
9.94 048 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

25 

26 

27 

28 
29 

9.69 010 
9.69 032 
9.69 055 
9.69 077 
9.69 100 

9.74 969 

9.74 998 

9.75 028 
9.75 058 
9.75 087 

10.25 031 
10.25 002 
10.24 972 
10.24 942 
10.24 913 

9.94 041 
9.94 034 
9.94 027 
9.94 020 
9.94 012 

40 

39 

38 

37 

36 

9.69 122 
9.69 144 
9.69 167 
9.69 189 
9.69 212 

9.75 117 
9.75 146 
9.75 176 
9.75 205 
9.75 235 

10.24 883 
10.24 854 
10.24 824 
10.24 795 
10.24 765 

9.94 005 
9.93 998 
9.93 991 
9.93 984 
9.93 977 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.69 234 
9.69 256 
9.69 279 
9.69 301 
9.69 323 

9.75 264 
9.75 294 
9.75 323 
9.75 353 
9.75 382 

10.24 736 
10.24 706 
10.24 677 
10.24 647 
10.24 618 

9.93 970 
9.93 963 
9.93 955 
9.93 948 
9.93 941 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.69 345 
9.69 368 
9.69 390 
9.69 412 
9.69 434 

9.75 411 
9.75 441 
9.75 470 
9.75 500 
9.75 529 

10.24 589 
10.24 559 
10.24 530 
10.24 500 
10.24 471 

9.93 934 
9.93 927 
9.93 920 
9.93 912 
9.93 905 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.69 456 
9.69 479 
9.69 501 
9.69 523 
9.69 545 

9.75 558 
9.75 588 
9.75 617 
9.75 647 
9.75 676 

10.24 442 
10.24 412 
10.24 383 
10.24 353 
10.24 324 

9.93 898 
9.93 891 
9.93 884 
9.93 876 
9.93 869 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.69 567 
9.69 589 
9.69 611 
9.69 633 
9.69 655 

9.75 705 
9.75 735 
9.75 764 
9.75 793 
9.75 822 

10.24 295 
10.24 265 
10.24 236 
10.24 207 
10.24 178 

9.93 862 
9.93 855 
9.93 847 
9.93 840 
9.93 833 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.69 677 
9.69 699 
9.69 721 
9.69 743 
9.69 765 

9.75 852 
9.75 881 
9.75 910 
9.75 939 
9.75 969 

10.24 148 
10.24 119 
10.24 090 
10.24 061 
10.24 031 

9.93 826 
9.93 819 
9.93 811 
9.93 804 
9.93 797 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

55 

56 

57 

58 

59 

9.69 787 
9.69 809 
9.69 831 
9.69 853 
9.69 875 

9.75 998 

9.76 027 
9.76 056 
9.76 086 
9.76 115 

10.24 002 
10.23 973 
10.23 944 
10.23 914 
10.23 885 

9.93 789 
9.93 782 
9.93 775 
9.93 768 
9.93 760 

60 

9.69 897 

9.76 144 

10.23 856 

9.93 753 


L Cos 

d 

L Cot 

cd 

L Tan 

L Sin 

d 

r 

Prop. Pts. 


60 ( 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


57 


30 ° 


t 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.69 897 


9.76 144 


10.23 856 

9.93 753 


60 

1 

9.69 919 

22 

9.76 173 

29 

10.23 827 

9.93 746 

7 

59 

2 

9.69 941 

22 

9.76 202 

29 

10.23 798 

9.93 738 

8 

58 

3 

9.69 963 

22 

9.76 231 

29 

10.23 769 

9.93 731 

7 

57 

4 

9.69 984 

21 

9.76 261 

30 

9Q 

10.23 739 

9.93 724 

7 

7 

56 

5 

9.70 006 


9.76 290 


10.23 710 

9.93 717 


55 

6 

9.70 028 

22 

9.76 319 

29 

10.23 681 

9.93 709 

8 

54 

7 

9.70 050 

22 

9.76 348 

29 

10.23 652 

9.93 702 

7 

53 

8 

9.70 072 

22 

9.76 377 

29 

10.23 623 

9.93 695 

7 

52 

9 

9.70 093 

21 

9.76 406 

29 

29 

10.23 594 

9.93 687 

8 

7 

51 

10 

9.70 115 


9.76 435 

10.23 565 

9.93 680 


50 

11 

9.70 137 

22 

9.76 464 

29 

10.23 536 

9.93 673 

7 

49 

12 

9.70 159 

22 

9.76 493 

29 

10.23 507 

9.93 665 

8 

48 

13 

9.70 180 

21 

9.76 522 

29 

10.23 478 

9.93 658 

7 

47 

14 

9.70 202 

22 

9.76 551 

29 

9Q 

10.23 449 

9.93 650 

8 

7 

46 

15 

9.70 224 


9.76 580 

Av 

10.23 420 

9.93 643 


45 

16 

9.70 245 

21 

9.76 609 

29 

10.23 391 

9.93 636 

7 

44 

17 

9.70 267 

22 

9.76 639 

30 

10.23 361 

9.93 628 

8 

43 

18 

9.70 288 

21 

9.76 668 

29 

10.23 332 

9.93 621 

7 

42 

19 

9.70 310 

22 

22 

9.76 697 

29 

90 

10.23 303 

9.93 614 

7 

8 

41 

20 

9.70 332 

9.76 725 

Zo 

10.23 275 

9.93 606 

40 

21 

9.70 353 

21 

9.76 754 

29 

10.23 246 

9.93 599 

7 

39 

22 

9.70 375 

22 

9.76 783 

29 

10.23 217 

9.93 591 

8 

38 

23 

9.70 396 

21 

9.76 812 

29 

10.23 188 

9.93 584 

7 

37 

24 

9.70 418 

22 

21 

9.76 841 

29 

90 

10.23 159 

9.93 577 

7 

8 

36 

25 

9.70 439 


9.76 870 


10.23 130 

9.93 569 


35 

26 

9.70 461 

22 

9.76 899 

29 

10.23 101 

9.93 562 

7 

34 

27 

9.70 482 

21 

9.76 928 

29 

10.23 072 

9.93 554 

8 

33 

28 

9.70 504 

22 

9.76 957 

29 

10.23 043 

9.93 547 

7 

32 

29 

9.70 525 

21 

oo 

9.76 986 

29 

90 

10.23 014 

9.93 539 

8 

7 

31 

30 

9.70 547 

LA 

9.77 015 

z \) 

10.22 985 

9.93 532 


30 

31 

9.70 568 

21 

9.77 044 

29 

10.22 956 

9.93 525 

7 

29 

32 

9.70 590 

22 

9.77 073 

29 

10.22 927 

9.93 517 

8 

28 

33 

9.70 611 

21 

9.77 101 

-28 

10.22 899 

9.93 510 

7 

27 

34 

9.70 633 

22 

9.77 130 

29 

90 

10.22 870 

9.93 502 

8 

7 

26 

35 

9.70 654 

21 

9.77 159 

zm 

10.22 841 

9.93 495 


25 

36 

9.70 675 

21 

9.77 188 

29 

10.22 812 

9.93 487 

8 

rr 

24 

37 

9.70 697 

22 

9.77 217 

29 

10.22 783 

9.93 480 

4 

Q 

23 

38 

9.70 718 

21 

9.77 246 

29 

10.22 754 

9.93 472 

«7 

22 

39 

9.70 739 

21 

9.77 274 

28 

9Q 

10.22 726 

9.93 465 

4 

8 

21 

40 

9.70 761 

22 

9.77 303 

Zi > 

10.22 697 

9.93 457 


20 

41 

9.70 782 

21 

9.77 332 

29 

10.22 668 

9.93 450 

7 

Q 

19 

42 

9.70 803 

21 

9.77 361 

29 

10.22 639 

9.93 442 

o 

n 

18 

43 

9.70 824 

21 

9.77 390 

29 

10.22 610 

9.93 435 

4 

Q 

17 

44 

9.70 846 

22 

9.77 418 

28 

9Q 

10.22 582 

9.93 427 

O 

7 

16 

45 

9.70 867 

21 

9.77 447 

Z \) 

10.22 553 

9.93 420 

Q 

15 

46 

9.70 888 

21 

9.77 476 

29 

10.22 524 

9.93 412 

o 

7 

14 

47 

9.70 909 

21 

9.77 505 

29 

10.22 495 

9.93 405 

4 

Q 

13 

48 

9.70 931 

22 

9.77 533 

28 

10.22 467 

9.93 397 

O 

n 

12 

49 

9.70 952 

21 

9.77 562 

29 

9Q 

10.22 438 

9.93 390 

4 

8 

11 

50 

9.70 973 

21 

9.77 591 

z\j 

10.22 409 

9.93 382 

7 

10 

51 

9.70 994 

21 

9.77 619 

28 

10.22 381 

9.93 375 

4 

g 

9 

52 

9.71 015 

21 

9.77 648 

29 

10.22 352 

9.93 367 

7 

8 

53 

9.71 036 

21 

9.77 677 

29 

10.22 323 

9.93 360 

Q 

7 

54 

9.71 058 

22 

9.77 706 

29 

9a 

10.22 294 

9.93 352 

O 

8 

6 

55 

9.71 079 

21 

9.77 734 

ZQ 

10.22 266 

9.93 344 

7 

5 

56 

9.71 100 

21 

9.77 763 

29 

10.22 237 

9.93 337 

4 

Q 

4 

57 

9.71 121 

21 

9.77 791 

28 

10.22 209 

9.93 329 

o 

7 

3 

58 

9.71 142 

21 

9.77 820 

29 

10.22 180 

9.93 322 

4 

g 

2 

59 

9.71 163 

21 

9.77 849 

29 

9a 

10.22 151 

9.93 314 

7 

1 

60 

9.71 184 

21 

9.77 877 

z o 

10.22 123 

9.93 307 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

r 


Prop. Pts. 



30 

29 

28 

6 

3.0 

2.9 

2.8 

7 

3.5 

3.4 

3.3 

8 

4.0 

3.9 

3.7 

9 

4.5 

4.4 

4.2 

10 

5.0 

4.8 

4.7 

20 

10.0 

9.7 

9.3 

30 

15.0 

14.5 

14.0 

40 

20.0 

19.3 

18.7 

50 

25.0 

24.2 

23.3 


" 

22 

21 

6 

2.2 

2.1 

7 

2.6 

2.4 

8 

2.9 

2.8 

9 

3.3 

3.2 

10 

3.7 

3.5 

20 

7.3 

7.0 

30 

11.0 

10.5 

40 

14.7 

14.0 

50 

18.3 

17.5 


Prop. Pts. 


59 ‘ 













































































58 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


31 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.71 184 
9.71 205 
9.71 226 
9.71 247 
9.71 268 

21 

21 

21 

21 

21 

21 

21 

21 

21 

20 

21 

21 

21 

21 

21 

21 

20 

21 

21 

21 

20 

21 

21 

21 

20 

21 

21 

20 

21 

21 

20 

21 

20 

21 

20 

21 

20 

21 

21 

20 

20 

21 

20 

21 

20 

21 

20 

20 

21 

20 

20 

21 

20 

20 

21 

20 

20 

21 

20 

20 

9.77 877 
9.77 906 
9.77 935 
9.77 963 
9.77 992 

29 

29 

28 

29 

28 

29 

28 

29 

29 

28 

29 

28 

29 

28 

29 

28 

29 

28 

28 

29 

28 

29 

28 

29 

28 

28 

29 

28 

29 

28 

28 

29 

28 

28 

29 

28 

28 

29 

28 

28 

28 

29 

28 

28 

28 

29 

28 

28 

28 

28 

29 

28 

28 

28 

28 

28 

29 

28 

28 

28 

10.22 123 
10.22 094 
10.22 065 
10.22 037 
10.22 008 

9.93 307 
9.93 299 
9.93 291 
9.93 284 
9.93 276 

8 

8 

7 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

7 

8 

8 

8 

7 

8 

8 

8 

7 

8 

8 

8 

8 

7 

8 

8 

8 

7 

8 

8 

8 

8 

8 

7 

8 

8 

8 

8 

8 

8 

7 

8 

8 

8 

8 

60 

59 

58 

57 

56 

" 29 28 

6 2.9 2.8 

7 3.4 3.3 

8 3.9 3.7 

9 4.4 4.2 

10 4.8 4.7 

20 9.7 9.3 

30 14.5 14.0 

40 19.3 18.7 

50 24.2 23.3 

" 21 20 

6 2.1 2.0 

7 2.4 2.3 

8 2.8 2.7 

9 3.2 3.0 

10 3.5 3.3 

20 7.0 6.7 

30 10.5 10.0 

40 14.0 13.3 

50 17.5 16.7 

"87 

6 0.8 0.7 

7 0.9 0.8 

8 1.1 0.9 

9 1.2 1.0 

10 1.3 1.2 

20 2.7 2.3 

30 4.0 3.5 

40 5.3 4.7 

50 6.7 5.8 

5 

6 

7 

8 

9 

9.71 289 
9.71 310 
9.71 331 
9.71 352 
9.71 373 

9.78 020 
9.78 049 
9.78 077 
9.78 106 
9.78 135 

10.21 980 
10.21 951 
10.21 923 
10.21 894 
10.21 865 

9.93 269 
9.93 261 
9.93 253 
9.93 246 
9.93 238 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.71 393 
9.71 414 
9.71 435 
9.71 456 
9.71 477 

9.78 163 
9.78 192 
9.78 220 
9.78 249 
9.78 277 

10.21 837 
10.21 808 
10.21 780 
10.21 751 
10.21 723 

9.93 230 
9.93 223 
9.93 215 
9.93 207 
9.93 200 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.71 498 
9.71 519 
9.71 539 
9.71 560 
9.71 581 

9.78 306 
9.78 334 
9.78 363 
9.78 391 
9.78 419 

10.21 694 
10.21 666 
10.21 637 
10.21 609 
10.21 581 

9.93 192 
9.93 184 
9.93 177 
9.93 169 
9.93 161 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.71 602 
9.71 622 
9.71 643 
9.71 664 
9.71 685 

9.78 448 
9.78 476 
9.78 505 
9.78 533 
9.78 562 

10.21 552 
10.21 524 
10.21 495 
10.21 467 
10.21 438 

9.93 154 
9.93 146 
9.93 138 
9.93 131 
9.93 123 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.71 705 
9.71 726 
9.71 747 
9.71 767 
9.71 788 

9.78 590 
9.78 618 
9.78 647 
9.78 675 
9.78 704 

10.21 410 
10.21 382 
10.21 353 
10.21 325 
10.21 296 

9.93 115 
9.93 108 
9.93 100 
9.93 092 
9.93 084 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.71 809 
9.71 829 
9.71 850 
9.71 870 
9.71 891 

9.78 732 
9.78 760 
9.78 789 
9.78 817 
9.78 845 

10.21 268 
10.21 240 
10.21 211 
10.21 183 
•10.21 155 

9.93 077 
9.93 069 
9.93 061 
9.93 053' 
9.93 046 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.71 911 
9.71 932 
9.71 952 
9.71 973 
9.71 994 

9.78 874 
9.78 902 
9.78 930 
9.78 959 
9.78 987 

10.21 126 
10.21 098 
10.21 070 
10.21 041 
10.21 013 

9.93 038 
9.93 030 
9.93 022 
9.93 014 
9.93 007 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.72 014 
9.72 034 
9.72 055 
9.72 075 
9.72 096 

9.79 015 
9.79 043 
9.79 072 
9.79 100 
9.79 128 

10.20 985 
10.20 957 
10.20 928 
10.20 900 
10.20 872 

9.92 999 
9.92 991 
9.92 983 
9.92 976 
9.92 968 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.72 116 
9.72 137 
9.72 157 
9.72 177 
9.72 198 

9.79 156 
9.79 185 
9.79 213 
9.79 241 
9.79 269 

10.20 844 
10.20 815 
10.20 787 
10.20 759 
10.20 731 

9.92 960 
9.92 952 
9.92 944 
9.92 936 
9.92 929 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.72 218 
9.72 238 
9.72 259 
9.72 279 
9.72 299 

9.79 297 
9.79 326 
9.79 354 
9.79 382 
9.79 410 

10.20 703 
10.20 674 
10.20 646 
10.20 618 
10.20 590 

9.92 921 
9.92 913 
9.92 905 
9.92 897 
9.92 889 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.72 320 
9.72 340 
9.72 360 
9.72 381 
9.72 401 

9.79 438 
9.79 466 
9.79 495 
9.79 523 
9.79 551 

10.20 562 
10.20 534 
10.20 505 
10.20 477 
10.20 449 

9.92 881 
9.92 874 
9.92 866 
9.92 858 
9.92 850 

5 

4 

3 

2 

1 

60 

9.72 421 

9.79 579 

10.20 421 

9.92 842 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. PtS. 


58 ‘ 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


59 


32 c 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.72 421 


9.79 579 


10.20 421 

9.92 842 


60 

1 

9.72 441 

20 

9.79 607 

28 

10.20 393 

9.92 834 

8 

59 

2 

9.72 461 

20 

9.79 635 

28 

10.20 365 

9.92 826 

8 

58 

3 

9.72 482 

21 

9.79 663 

28 

10.20 337 

9.92 818 

8 

57 

4 

9.72 502 

20 

20 

9.79 691 

28 

28 

10.20 309 

9.92 810 

8 

56 

5 

9.72 522 

9.79 719 

10.20 281 

9.92 803 

7 


6 

9.72 542 

20 

9.79 747 

28 

10.20 253 

9.92 795 

8 

54 

7 

9.72 562 

20 

9.79 776 

29 

10.20 224 

9.92 787 

8 

53 

8 

9.72 582 

20 

9.79 804 

28 

10.20 196 

9.92 779 

8 

52 

9 

9.72 602 

20 

20 

9.79 832 

28 

28 

10.20 168 

9.92 771 

8 

51 

10 

9.72 622 

9.79 860 

10.20 140 

9.92 763 

8 

50 

11 

9.72 643 

21 

9.79 888 

28 

10.20 112 

9.92 755 

8 

49 

12 

9.72 663 

20 

9.79 916 

28 

10.20 084 

9.92 747 

8 

48 

13 

9.72 683 

20 

9.79 944 

28 

10.20 056 

9.92 739 

8 

47 

14 

9.72 703 

20 

20 

9.79 972 

28 

28 

10.20 028 

9.92 731 

8 

O 

46 

15 

9.72 723 

9.80 000 

10.20 000 

9.92 723 

0 

45 

16 

9.72 743 

20 

9.80 028 

28 

10.19 972 

9.92 715 

8 

44 

17 

9.72 763 

20 

9.80 056 

28 

10.19 944 

9.92 707 

8 

43 

18 

9.72 783 

20 

9.80 084 

28 

10.19 916 

9.92 699 

8 

42 

19 

9.72 803 

20 

20 

9.80 112 

28 

28 

10.19 888 

9.92 691 

8 

Q 

41 

20 

9.72 823 

9.80 140 

10.19 860 

9.92 683 

O 

40 

21 

9.72 843 

20 

9.80 168 

28 

10.19 832 

9.92 675 

8 

39 

22 

9.72 863 

20 

9.80 195 

27 

10.19 805 

9.92 667 

8 

38 

23 

9.72 883 

20 

9.80 223 

28 

10.19 777 

9.92 659 

8 

37 

24 

9.72 902 

19 

20 

9.80 251 

28 

28 

10.19 749 

9.92 651 

8 

8 

36 

25 

9.72 922 

9.80 279 

10.19 721 

9.92 643 

35 

26 

9.72 942 

20 

9.80 307 

28 

10.19 693 

9.92 635 

8 

34 

27 

9.72 962 

20 

9.80 335 

28 

10.19 665 

9.92 627 

8 

33 

28 

9.72 982 

20 

9.80 363 

28 

10.19 637 

9.92 619 

8 

32 

29 

9.73 002 

20 

20 

9.80 391 

28 

28 

10.19 609 

9.92 611 

8 

8 

31 

30 

9.73 022 

9.80 419 

10.19 581 

9.92 603 

30 

31 

9.73 041 

19 

9.80 447 

28 

10.19 553 

9.92 595 

8 

29 

32 

9.73 061 

20 

9.80 474 

27 

10.19 526 

9.92 587 

8 

28 

33 

9.73 081 

20 

9.80 502 

28 

10.19 498 

9.92 579 

8 

27 

34 

9.73 101 

20 

Oft 

9.80 530 

28 

98 

10.19 470 

9.92 571 

8 

g 

26 

35 

9.73 121 


9.80 558 


10.19 442 

9.92 563 


25 

36 

9.73 140 

19 

9.80 586 

28 

10.19 414 

9.92 555 

8 

24 

37 

9.73 160 

20 

9.80 614 

28 

10.19 386 

9.92 546 

9 

23 

38 

9.73 180 

20 

9.80 642 

28 

10.19 358 

9.92 538 

8 

22 

39 

9.73 200 

20 

1 Q 

9.80 669 

27 

98 

10.19 331 

9.92 530 

8 

g 

21 

40 

9.73 219 


9.80 697 

60 

10.19 303 

9.92 522 


20 

41 

9.73 239 

20 

9.80 725 

28 

10.19 275 

9.92 514 

8 

19 

42 

9.73 259 

20 

9.80 753 

28 

10.19 247 

9.92 506 

8 

18 

43 

9.73 278 

19 

9.80 781 

28 

10.19 219 

9.92 498 

8 

17 

44 

9.73 298 

20 

on 

9.80 808 

27 

98 

10.19 192 

9.92 490 

8 

g 

16 

45 

9.73 318 


9.80 836 

60 

10.19 164 

9.92 482 


15 

46 

9.73 337 

19 

9.80 864 

28 

10.19 136 

9.92 473 

9 

14 

47 

9.73 357 

20 

9.80 892 

28 

10.19 108 

9.92 465 

8 

13 

48 

9.73 377 

20 

9.80 919 

27 

10.19 081 

9.92 457 

8 

12 

49 

9.73 396 

19 

on 

9.80 947 

28 

98 

10.19 053 

9.92 449 

8 

8 

11 

50 

9.73 416 


9.80 975 


10.19 025 

9.92 441 

10 

51 

9.73 435 

19 

9.81 003 

28 

10.18 997 

9.92 433 

8 

9 

52 

9.73 455 

20 

9.81 030 

27 

10.18 970 

9.92 425 

8 

8 

53 

9.73 474 

19 

9.81 058 

28 

10.18 942 

9.92 416 

9 

7 

54 

9.73 494 

20 

9.81 086 

28 

97 

10.18 914 

9.92 408 

8 

g 

6 

55 

9.73 513 

19 

9.81 113 

6 l 

10.18 887 

9.92 400 


5 

56 

9.73 533 

20 

9.81 141 

28 

10.18 859 

9.92 392 

8 

4 

57 

9.73 552 

19 

9.81 169 

28 

10.18 831 

9.92 384 

8 

3 

58 

9.73 572 

20 . 

9.81 196 

27 

10.18 804 

9.92 376 

8 

2 

59 

9.73 591 

19 

on 

9.81 224 

28 

98 

10.18 776 

9.92 367 

9 

g 

1 

60 

9.73 611 


9.81 252 

60 

10.18 748 

9.92 359 


0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 


Prop. Pts. 


" 

29 

28 

27 

6 

2.9 

2.8 

2.7 

7 

3.5 

3.3 

3.2 

8 

3.9 

3.7 

3.6 

9 

4.4 

4.2 

4.0 

10 

4.8 

4.7 

4.5 

20 

9.7 

9.3 

9.0 

30 

14.5 

14.0 

13.5 

40 

19.3 

18.7 

18.0 

50 

24.2 

23.3 

22.5 


" 

21 

20 

6 

2.1 

2.0 

7 

2.4 

2.3 

8 

2.8 

2.7 

9 

3.2 

3.0 

10 

3.5 

3.3 

20 

7.0 

6.7 

30 

10.5 

10.0 

40 

14.0 

13.3 

50 

17.5 

16.7 


19 

1.9 

2.2 

2.5 
2.8 

3.2 

6.3 

9.5 

12.7 

15.8 


" 

9 

8 

6 

0.9 

0.8 

7 

1.0 

0.9 

8 

1.2 

1.1 

9 

1.4 

1.2 

10 

1.5 

1.3 

20 

3.0 

2.7 

30 

4.5 

4.0 

40 

6.0 

5.3 

50 

7.5 

6.7 


Prop. Pts. 


57 ' 







































































60 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


33° 


/ 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.73 611 
9.73 630 
9.73 650 
9.73 669 
9.73 689 

19 

20 

19 

20 
19 

19 

20 
19 

19 

20 

19 

19 

20 
19 

19 

20 
19 
19 
19 

19 

20 
19 
19 
19 

19 

20 
19 
19 
19 
19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

19 

18 

19 

19 

19 

19 

19 

18 

19 

19 

19 

18 

19 

9.81 252 
9.81 279 
9.81 307 
9.81 335 
9.81 362 

27 

28 
28 

27 

28 

28 

27 

28 

27 

28 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 

27 

28 
27 

27 

28 

27 

28 
27 

27 

28 

27 

28 
27 

27 

28 
27 

27 

28 
27 

27 

28 
27 
27 

27 

28 
27 
27 

27 

28 

10.18 748 
10.18 721 
10.18 693 
10.18 665 
10.18 638 

9.92 359 
9.92 351 
9.92 343 
9.92 335 
9.92 326 

8 

8 

8 

9 

8 

8 

8 

9 

8 

8 

8 

9 

8 

8 

9 

8 

8 

8 

9 

8 

8 

9 

8 

8 

9 

8 

8 

9 

8 

8 

9 

8 

8 

9 

8 

9 

8 

8 

9 

-8 

9 

8 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

60 

59 

58 

57 

56 

" 28 27 20 

6 2.8 2.7 2.0 

7 3.3 3.2 2.3 

8 3.7 3.6 2.7 

9 4.2 4.0 3.0 

10 4.7 4.5 3.3 

20 9.3 9.0 6.7 

30 14.0 13.5 10.0 

40 18.7 18.0 13.3 

50 23.3 22.5 16.7 

" 19 18 

6 1.9 1.8 

7 2.2 2.1 

8 2.5 2.4 

9 2.9 2.7 

10 3.2 3.0 

20 6.3 6.0 

30 9.5 9.0 

40 12.7 12.0 

50 15.8 15.0 

"98 

6 0.9 0.8 

7 1.0 0.9 

8 1.2 1.1 

9 1.4 1.2 

10 1.5 1.3 

20 3.0 2.7 

30 4.5 4.0 

40 6.0 5.3 

50 7.5 6.7 

5 

6 

7 

8 

9 

9.73 708 
9.73 727 
9.73 747 
9.73 766 
9.73 785 

9.81 390 
9.81 418 
9.81 445 
9.81 473 
9.81 500 

10.18 610 
10.18 582 
10.18 555 
10.18 527 
10.18 500 

9.92 318 
9.92 310 
9.92 302 
9.92 293 
9.92 285 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.73 805 
9.73 824 
9.73 843 
9.73 863 
9.73 882 

9.81 528 
9.81 556 
9.81 583 
9.81 611 
9.81 638 

10.18 472, 
10.18 444 
10.18 417 
10.18 389 
10.18 362 

9.92 277 
9.92 269 
9.92 260 
9.92 252 
9.92 244 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.73 901 
9.73 921 
9.73 940 
9.73 959 
9.73 978 

9.81 666 
9.81 693 
9.81 721 
9.81 748 
9.81 776 

10.18 334 
10.18 307 
10.18 279 
10.18 252 
10.18 224 

9.92 235 
9.92 227 
9.92 219 
9.92 211 
9.92 202 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.73 997 

9.74 017 
9.74 036 
9.74 055 
9.74 074 

9.81 803 
9.81 831 
9.81 858 
9.81 886 
9.81 913 

10.18 197 
10.18 169 
10.18 142 
10.18 114 
10.18 087 

9.92 194 
9.92 186 
9.92 177 
9.92 169 
9.92 161 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.74 093 
9.74 113 
9.74 132 
9.74 151 
9.74 170 

9.81 941 
9.81 968 

9.81 996 

9.82 023 
9.82 051 

10.18 059 
10.18 032 
10.18 004 
10.17 977 
10.17 949 

9.92 152 
9.92 144 
9.92 136 
9.92 127 
9.92 119 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.74 189 
9.74 208 
9.74 227 
9.74 246 
9.74 265 

9.82 078 
9.82 106 
9.82 133 
9.82 161 
9.82 188 

10.17 922 
10.17 894 
10.17 867 
10.17 839 
10.17 812 

9.92 111 
9.92 102 
9.92 094 
9.92 086 
9.92 077 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.74 284 
9.74 303 
9.74 322 
9.74 341 
9.74 360 

9.82 215 
9.82 243 
9.82 270 
9.82 298 
9.82 325 

10.17 785 
10.17 757 
10.17 730 
10.17 702 
10.17 675 

9.92 069 
9.92 060 
9.92 052 
9.92 044 
9.92 035 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.74 379 
9.74 398 
9.74 417 
9.74 436 
9.74 455 

9.82 352 
9.82 380 
9.82 407 
9.82 435 
9.82 462 

10.17 648 
10.17 620 
10.17 593 
10.17 565 
10.17 538 

9.92 027 
9.92 018 
9.92 010 
9.92 002 
9.91 993 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.74 474 
9.74 493 
9.74 512 
9.74 531 
9.74 549 

9.82 489 
9.82 517 
9.82 544 
9.82 571 
9.82 599 

10.17 511 
10.17 483 
10.17 456 
10.17 429 
10.17 401 

9.91 985 
9.91 976 
9.91 968 
9.91 959 
9.91 951 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.74 568 
9.74 587 
9.74 606 
9.74 625 
9.74 644 

9.82 626 
9.82 653 
9.82 681 
9.82 708 
9.82 735 

10.17 374 
10.17 347 
10.17 319 
10.17 292 
10.17 265 

9.91 942 
9.91 934 
9.91 925 
9.91 917 
9.91 908 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.74 662 
9.74 681 
9.74 700 
9.74 719 
9.74 737 

9.82 762 
9.82 790 
9.82 817 
9.82 844 
9.82 871 

10.17 238 
10.17 210 
10.17 183 
10.17 156 
10.17 129 

9.91 900 
9.91 891 
9.91 883 
9.91 874 
9.91 866 

5 

4 

3 

2 

1 

60 

9.74 756 

9.82 899 

10.17 101 

9.91 857 

0 


L Cos 

d 

L Cot 

c d 

L Tan | L Sin 

d 

' 

Prop. Pts. 


56 







































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


61 


34° 


7 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

9.74 756 

9.74 775 
9.74 794 
9.74 812 
9.74 831 

19 

19 

18 

19 

19 

18 

19 

19 

18 

19 

18 

19 

19 

18 

19 

18 

19 

18 

19 

18 

19 

18 

19 

18 

19 

18 

19 

18 

18 

19 

18 

19 

18 

18 

19 

18 

18 

18 

19 

18 

18 

19 

18 

18 

18 

18 

19 

18 

18 

18 

18 

18 

19 

18 

18 

18 

18 

18 

18 

18 

9.82 899 
9.82 926 
9.82 953 

9.82 980 

9.83 008 

27 

27 

27 

28 
27 

27 

27 

28 
27 
27 

27 

27 

27 

28 
27 

27 

27 

27 

27 

27 

28 
27 
27 
27 
27 

27 

27 

27 

27 

27 

27 

28 
27 
27 
27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

26 

27 

27 

27 

27 

27 

27 

27 

27 

27 

10.17 101 
10.17 074 
10.17 047 
10.17 020 
10.16 992 

9.91 857 
9.91 849 
9.91 840 
9.91 832 
9.91 823 

8 

9 

8 

9 

8 

9 

8 

9 

8 

9 

9 

8 

9 

8 

9 

9 

8 

9 

8 

9 

8 

9 

9 

8 

9 

9 

8 

9 

9 

8 

9 

9 

8 

9 

9 

9 

8 

9 

9 

8 

9 

9 

9 

8 

9 

9 

9 

9 

8 

9 

9 

9 

9 

8 

9 

9 

9 

9 

9 

60 

59 

58 

57 

56 

" 28 27 26 

6 2.8 2.7 2.6 

7 3.3 3.2 3.0 

8 3.7 3.6 3.5 

9 4.2 4.0 3.9 

10 4.7 4.5 4.3 

20 9.3 9.0 8.7 

30 14.0 13.5 13.0 

40 18.7 18.0 17.3 

50 23.3 22.5 21.7 

" 19 18 

6 1.9 1.8 

7 2.2 2.1 

8 2.5 2.4 

9 2.8 2.7 

10 3.2 3.0 

20 6.3 6.0 

30 9.5 9.0 

40 12.7 12.0 

50 15.8 15.0 

"98 

6 0.9 0.8 

7 1.0 0.9 

8 1.2 1.1 

9 1.4 1.2 

10 1.5 1.3 

20 3.0 2.7 

30 4.5 4.0 

40 6.0 5.3 

50 7.5 6.7 

9.74 850 
9.74 868 
9.74 887 
9.74 906 
9.74 924 

9.74 943 
9.74 961 
9.74 980 

9.74 999 

9.75 017 

9.83 035 
9.83 062 
9.83 089 
9.83 117 
9.83 144 

10.16 965 
10.16 938 
10.16 911 
10.16 883 
10.16 856 

9.91 815 
9.91 806 
9.91 798 
9.91 789 
9.91 781 

55 

54 

53 

52 

51 

9.83 171 
9.83 198 
9.83 225 
9.83 252 
9.83 280 

10.16 829 
10.16 802 
10.16 775 
10.16 748 
10.16 720 

9.91 772 
9.91 763 
9.91 755 
9.91 746 
9.91 738 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.75 036 
9.75 054 
9.75 073 
9.75 091 
9.75 110 

9.83 307 
9.83 334 
9.83 361 
9.83 388 
9.83 415 

10.16 693 
10.16 666 
10.16 639 
10.16 612 
10.16 585 

9.91 729 
9.91 720 
9.91 712 
9.91 703 
9.91 695 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.75 128 
9.75 147 
9.75 165 
9.75 184 
9.75 202 

9.83 442 
9.83 470 
9.83 497 
9.83 524 
9.83 551 

10.16 558 
10.16 530 
10.16 503 
10.16 476 
10.16 449 

9.91 686 
9.91 677 
9.91 669 
9.91 660 
9.91 651 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.75 221 
9.75 239 
9.75 258 
9.75 276 
9.75 294 

9.83 578 
9.83 605 
9.83 632 
9.83 659 
9.83 686 

10.16 422 
10.16 395 
10.16 368 
10.16 341 
10.16 314 

9.91 643 
9.91 634 
9.91 625 
9.91 617 
9.91 608 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.75 313 
9.75 331 
9.75 350 
9.75 368 
9.75 386 

9.83 713 
9.83 740 
9.83 768 
9.83 795 
9.83 822 

10.16 287 
10.16 260 
10.16 232 
10.16 205 
10.16 178 

9.91 599 
9.91 591 
9.91 582 
9.91 573 
9.91 565 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.75 405 
9.75 423 
9.75 441 
9.75 459 
9.75 478 

9.83 849 
9.83 876 
9.83 903 
9.83 930 
9.83 957 

10.16 151 
10.16 124 
10.16 097 
10.16 070 
10.16 043 

9.91 556 
9.91 547 
9.91 538 
9.91 530 
9.91 521 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.75 496 
9.75 514 
9.75 533 
9.75 551 
9.75 569 

9.83 984 

9.84 011 
9.84 038 
9.84 065 
9.84 092 

10.16 016 
10.15 989 
10.15 962 
10.15 935 
10.15 908 

9.91 512 
9.91 504 
9.91 495 
9.91 486 
9.91 477 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.75 587 
9.75 605 
9.75 624 
9.75 642 
9.75 660 

9.84 119 
9.84 146 
9.84 173 
9.84 200 
9.84 227 

10.15 881 
10.15 854 
10.15 827 
10.15 800 
10.15 773 

9.91 469 
9.91 460 
9.91 451 
9.91442 
9.91 433 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.75 678 
9.75 696 
9.75 714 
9.75 733 
9.75 751 

9.84 254 
9.84 280 
9.84 307 
9.84 334 
9.84 361 

10.15 746 
10.15 720 
10.15 693 
10.15 666 
10.15 639 

9.91 425 
9.91 416 
9.91 407 
9.91 398 
9.91 389 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.75 769 
9.75 787 
9.75 805 
9.75 823 
9.75 841 

9.84 388 
9.84 415 
9.84 442 
9.84 469 
9.84 496 

10.15 612 
10.15 585 
10.15 558 
10.15 531 
10.15 504 

9.91 381 
9.91 372 
9.91 363 
9.91 354 
9.91 345 

5 

4 

3 

2 

1 

60 

9.75 859 

9.84 523 

10.15 477 

9.91 336 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


55 









































































62 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


35 c 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


0 

9.75 859 

18 

9.84 523 

27 

10.15 

47. 

9.91 336 


60 

1 

9.75 877 

9.84 550 

10.15 

450 

9.91 328 

8 

59 

2 

9.75 895 

18 

9.84 576 

26 

10.15 

424 

9.91 319 

9 

58 

3 

9.75 913 

18 

9.84 603 

27 

10.15 

397 

9.91 310 

9 

57 

4 

9.75 931 

18 

9.84 630 

27 

07 

10.15 

370 

9.91 301 

9 

9 

56 

5 

9.75 949 


9.84 657 

Z { 

10.15 

343 

9.91 292 


55 

6 

9.75 967 

18 

9.84 684 

27 

10.15 

316 

9.91 283 

9 

54 

7 

9.75 985 

18 

9.84 711 

27 

10.15 

289 

9.91 274 

9 

53 

8 

9.76 003 

18 

9.84 738 

27 

10.15 

262 

9.91 266 

8 

52 

9 

9.76 021 

18 

9.84 764 

26 

97 

10.15 

236 

9.91 257 

9 

9 

51 

10 

9.76 039 


9.84 791 


10.15 

209 

9.91 248 


50 

11 

9.76 057 

18 

9.84 818 

27 

10.15 

182 

9.91 239 

9 

49 

12 

9.76 075 

18 

9.84 845 

27 

10.15 

155 

9.91 230 

9 

48 

13 

9.76 093 

18 

9.84 872 

27 

10.15 

128 

9.91 221 

9 

47 

14 

9.76 111 

18 

9.84 899 

27 

Oft 

10.15 

101 

9.91 212 

9 

9 

46 

15 

9.76 129 


9.84 925 

zu 

10.15 

075 

9.91 203 


45 

16 

9.76 146 

17 

9.84 952 

27 

10.15 

048 

9.91 194 

9 

44 

17 

9.76 164 

18 

9.84 979 

27 

10.15 

021 

9.91 185 

9 

43 

18 

9.76 182 

18 

9.85 006 

27 

10.14 

994 

9.91 176 

9 

42 

19 

9.76 200 

18 

9.85 033 

27 

Of! 

10.14 

967 

9.91 167 

9 

9 

41 

20 

9.76 218 


9.85 059 

ZO 

10.14 

941 

9.91 158 


40 

21 

9.76 236 

18 

9.85 086 

27 

10.14 

914 

9.91 149 

9 

39 

22 

9.76 253 

17 

9.85 113 

27 

10.14 

887 

9.91 141 

8 

38 

23 

9.76 271 

18 

9.85 140 

27 

10.14 

860 

9.91 132 

9 

37 

24 

9.76 289 

18 
i a 

9.85 166 

26 

97 

10.14 

834 

9.91 123 

9 

9 

36 

25 

9.76 307 

1 o 

9.85 193 

Z i 

10.14 

807 

9.91 114 


35 

26 

9.76 324 

17 

9.85 220 

27 

10.14 

780 

9.91 105 

9 

34 

27 

9.76 342 

18 

9.85 247 

27 

10.14 

753 

9.91 096 

9 

33 

28 

9.76 360 

18 

9.85 273 

26 

10.14 

727 

9.91 087 

9 

32 

29 

9.76 378 

18 

1 7 

9.85 300 

27 

97 

10.14 

700 

9.91 078 

9 

9 

31 

30 

9.76 395 

i -1 

9.85 327 


10.14 

673 

9.91 069 


30 

31 

9.76 413 

18 

9.85 354 

27 

10.14 

646 

9.91 060 

9 

29 

32 

9.76 431 

18 

9.85 380 

26 

10.14 

620 

9.91 051 

9 

28 

33 

9.76 448 

17 

9.85 407 

27 

10.14 

593 

9.91 042 

9 

27 

34 

9.76 466 

18 
i a 

9.85 434 

27 

9ft 

10.14 

566 

9.91 033 

9 

10 

26 

35 

9.76 484 

±o 

9.85 460 


10.14 

540 

9.91 023 


25 

36 

9.76 501 

17 

9.85 487 

27 

10.14 

513 

9.91 014 

9 

24 

37 

9.76 519 

18 

9.85 514 

27 

10.14 

486 

9.91 005 

9 

23 

38 

9.76 537 

18 

9.85 540 

26 

10.14 

460 

9.90 996 

9 

22 

39 

9.76 554 

17 
i a 

9.85 567 

27 

97 

10.14 

433 

9.90 987 

9 

9 

21 

40 

9.76 572 

lo 

9.85 594 

Z t 

10.14 

406 

9.90 978 


20 

41 

9.76 590 

18 

9.85 620 

26 

10.14 

380 

9.90 969 

9 

19 

42 

9.76 607 

17 

9.85 647 

27 

10.14 

353 

9.90 960 

9 

18 

43 

9.76 625 

18 

9.85 674 

27 

10.14 

326 

9.90 951 

9 

17 

44 

9.76 642 

17 
i a 

9.85 700 

26 

27 

10.14 

300 

9.90 942 

9 

9 

16 

45 

9.76 660 

lo 

9.85 727 

10.14 

273 

9.90 933 


15 

46 

9.76 677 

17 

9.85 754 

27 

10.14 

246 

9.90 924 

9 

14 

47 

9.76 695 

18 

9.85 780 

26 

10.14 

220 

9.90 915 

9 

13 

48 

9.76 712 

17 

9.85 807 

27 

10.14 

193 

9.90 906 

9 

12 

49 

9.76 730 

18 

1 7 

9.85 834 

27 

9fi 

10.14 

166 

9.90 896 

10 

9 

11 

50 

9.76 747 


9.85 860 

ZD 

10.14 

140 

9.90 887 

9 

10 

51 

9.76 765 

18 

9.85 887 

27 

10.14 

113 

9.90 878 

9 

52 

9.76 782 

17 

9.85 913 

26 

10.14 

087 

9.90 869 

9 

8 

53 

9.76 800 

18 

9.85 940 

27 

10.14 

060 

9.90 860 

9 

7 

54 

9.76 817 

17 
i a 

9.85 967 

27 

26 

10.14 

033 

9.90 851 

9 

9 

6 

55 

9.76 835 

lo 

9.85 993 

10.14 

007 

9.90 842 

10 

5 

56 

9.76 852 

17 

9.86 020 

27 

10.13 

980 

9.90 832 

4 

57 

9.76 870 

18 

9.86 046 

26 

10.13 

954 

9.90 823 

9 

3 

58 

9.76 887 

17 

9.86 073 

27 

10.13 

927 

9.90 814 

9 

2 

59 

9.76 904 

17 

18 

9.86 100 

27 

26 

10.13 

900 

9.90 805 

9 

9 

1 

60 

9.76 922 

9.86 126 

10.13 

00 

9.90 796 


0 


L Cos 

d 

L Cot 

cd 

L Tan 

L Sin 

d 

' 


Prop. Pts. 


" 

27 

26 

18 

6 

2.7 

2.6 

1.8 

7 

3.2 

3.0 

2.1 

8 

3.6 

3.5 

2.4 

9 

4.0 

3.9 

2.7 

10 

4.5 

4.3 

3.0 

20 

9.0 

8.7 

6.0 

30 

13.5 

13.0 

9.0 

40 

18.0 

17.3 

12.0 

50 

22.5 

21.7 

15.0 


Prop. Pts. 


54 ( 











































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


63 


36° 


r 

L Sin 

d J 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.76 922 
9.76 939 
9.76 957 
9.76 974 
9.76 991 

17 

18 
17 

17 

18 

17 

17 

18 
17 
17 

17 

18 
17 
17 

17 

18 
17 
17 

17 

18 

17 

17 

17 

17 

17 

17 

17 

18 
17 
17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

17 

16 

17 

17 

17 

17 

17 

17 

16 

17 

17 

17 

17 

16 

9.86 126 
9.86 153 
9.86 179 
9.86 206 
9.86 232 

27 

26 

27 

26 

27 

26 

27 

26 

27 

27 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

26 

27 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

26 

10.13 874 
10.13 847 
10.13 821 
10.13 794 
10.13 768 

9.90 796 
9.90 787 
9.90 777 
9.90 768 
9.90 759 

9 

10 

9 

9 

9 

9 

10 

9 

9 

9 

10 

9 

9 

9 

10 

9 

9 

9 

10 

9 

9 

10 

9 

9 

9 

10 

9 

9 

10 

9 

9 
10 

9 

10 

9 

9 

10 

9 

9 

10 

9 

10 

9 

10 

9 

9 

10 

9 

10 

9 

10 

9 

10 

9 

10 

9 

10 

9 

10 

9 

60 

59 

58 

57 

56 

" 27 26 18 

6 2.7 2.6 1.8 

7 3.2 3.0 2.1 

8 3.6 3.5 2.4 

9 4.0 3.9 2.7 

10 4.5 4.3 3.0 

20 9.0 8.7 6.0 

30 13.5 13.0 9.0 

40 18.0 17.3 12.0 

50 22.5 21.7 15.0 

" 17 16 

6 1.7 1.6 

7 2.0 1.9 

8 2.3 2.1 

9 2.6 2.4 

10 2.8 2.7 

20 5.7 5.3 

30 8.5 8.0 

40 11.3 10.7 

50 14.2 13.3 

" 10 9 

6 1.0 0.9 

7 1.2 1.0 

8 1.3 1.2 

9 1.5 1.4 

10 1.7 1.5 

20 3.3 3.0 

30 5.0 4.5 

40 6.7 6.0 

50 8.3 7.5 

5 

6 

7 

8 

9 

9.77 009 
9.77 026 
9.77 043 
9.77 061 
9.77 078 

9.86 259 
9.86 285 
9.86 312 
9.86 338 
9.86 365 

10.13 741 
10.13 715 
10.13 688 
10.13 662 
10.13 635 

9.90 750 
9.90 741 
9.90 731 
9.90 722 
9.90 713 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.77 095 
9.77 112 
9.77 130 
9.77 147 
9.77 164 

9.86 392 
9.86 418 
9.86 445 
9.86 471 
9.86 498 

10.13 608 
10.13 582 
10.13 555 
10.13 529 
10.13 502 

9.90 704 
9.90 694 
9.90 685 
9.90 676 
9.90 667 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.77 181 
9.77 199 
9.77 216 
9.77 233 
9.77 250 

9.86 524 
9.86 551 
9.86 577 
9.86 603 
9.86 630 

10.13 476 
10.13 449 
10.13 423 
10.13 397 
10.13 370 

9.90 657 
9.90 648 
9.90 639 
9.90 630 
9.90 620 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.77 268 
9.77 285 
9.77 302 
9.77 319 
9.77 336 

9.86 656 
9.86 683 
9.86 709 
9.86 736 
9.86 762 

10.13 344 
10.13 317 
10.13 291 
10.13 264 
10.13 238 

9.90 611 
9.90 602 
9.90 592 
9.90 583 
9.90 574 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.77 353 
9.77 370 
9.77 387 
9.77 405 
9.77 422 

9.86 789 
9.86 815 
9.86 842 
9.86 868 
9.86 894 

10.13 211 
10.13 185 
10.13 158 
10.13 132 
10.13 106 

9.90 565 
9.90 555 
9.90 546 
9.90 537 
9.90 527 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.77 439 
9.77 456 
9.77 473 
9.77 490 
9.77 507 

9.86 921 
9.86 947 

9.86 974 

9.87 000 
9.87 027 

10.13 079 
10.13 053 
10.13 026 
10.13 000 
10.12 973 

9.90 518 
9.90 509 
9.90 499 
9.90 490 
9.90 480 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.77 524 
9.77 541 
9.77 558 
9.77 575 
9.77 592 

9.87 053 
9.87 079 
9.87 106 
9.87 132 
9.87 158 

10.12 947 
10.12 921 
10.12 894 
10.12 868 
10.12 842 

9.90 471 
9.90 462 
9.90 452 
9.90 443 
9.90 434 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.77 609 
9.77 626 
9.77 643 
9.77 660 
9.77 677 

9.87 185 
9.87 211 
9.87 238 
9.87 264 
9.87 290 

10.12 815 
10.12 789 
10.12 762 
10.12 736 
10.12 710 

9.90 424 
9.90 415 
9.90 405 
9.90 396 
9.90 386 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.77 694 
9.77 711 
9.77 728 
9.77 744 
9.77 761 

9.87 317 
9.87 343 
9.87 369 
9.87 396 
9.87 422 

10.12 683 
10.12 657 
10.12 631 
10.12 604 
10.12 578 

9.90 377 
9.90 368 
9.90 358 
9.90 349 
9.90 339 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.77 778 
9.77 795 
9.77 812 
9.77 829 
9.77 846 

9.87 448 
9.87 475 
9.87 501 
9.87 527 
9.87 554 

10.12 552 
10.12 525 
10.12 499 
10.12 473 
10.12 446 

9.90 330 
9.90 320 
9.90 311 
9.90 301 
9.90 292 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.77 862 
9.77 879 
9.77 896 
9.77 913 
9.77 930 

9.87 580 
9.87 606 
9.87 633 
9.87 659 
9.87 685 

10.12 420 
10.12 394 
10.12 367 
10.12 341 
10.12 315 

9.90 282 
9.90 273 
9.90 263 
9.90 254 
9.90 244 

5 

4 

3 

2 

1 

60 

9.77 946 

9.87 711 

10.12 289 

9.90 235 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


53 











































































64 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


37° 


' 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 

d 


0 

9.77 946 


9.87 711 


10.12 289 

9.90 235 


60 

1 

9.77 963 

17 

9.87 738 

27 

10.12 262 

9.90 225 

10 

59 

2 

9.77 980 

17 

9.87 764 

26 

10.12 236 

9.90 216 

9 

58 

3 

9.77 997 

17 

9.87 790 

26 

10.12 210 

9.90 206 

10 

57 

4 

9.78 013 

16 

17 

9.87 817 

27 

o A 

10.12 183 

9.90 197 

9 

10 

56 

5 

9.78 030 

9.87 843 

ZO 

10.12 157 

9.90 187 

55 

6 

9.78 047 

17 

9.87 869 

26 

10.12 131 

9.90 178 

9 

54 

7 

9.78 063 

16 

9.87 895 

26 

10.12 105 

9.90 168 

10 

53 

8 

9.78 080 

17 

9.87 922 

27 

10.12 078 

9.90 159 

9 

52 

9 

9.78 097 

17 

9.87 948 

26 

9A 

10.12 052 

9.90 149 

10 

10 

51 

10 

9.78 113 


9.87 974 


10.12 026 

9.90 139 

50 

11 

9.78 130 

17 

9.88 000 

26 

10.12 000 

9.90 130 

9 

49 

12 

9.78 147 

17 

9.88 027 

27 

10.11 973 

9.90 120 

10 

48 

13 

9.78 163 

16 

9.88 053 

26 

10.11 947 

9.90 111 

9 

47 

14 

9.78 180 

17 

9.88 079 

26 

10.11 921 

9.90 101 

10 

46 





26 



10 


15 

9.78 197 


9.88 105 

10.11 895 

9.90 091 

45 

16 

9.78 213 

16 

9.88 131 

26 

10.11 869 

9.90 082 

9 

44 

17 

9.78 230 

17 

9.88 158 

27 

10.11 842 

9.90 072 

10 

43 

18 

9.78 246 

16 

9.88 184 

26 

10.11 816 

9.90 063 

9 

42 

19 

9.78 263 

17 

9.88 210 

26 

9A 

10.11 790 

9.90 053 

10 

10 

41 

20 

9.78 280 


9.88 236 

ZO 

10.11 764 

9.90 043 

40 

21 

9.78 296 

16 

9.88 262 

26 

10.11 738 

9.90 034 

9 

39 

22 

9.78 313 

17 

9.88 289 

27 

10.11 711 

9.90 024 

10 

38 

23 

9.78 329 

16 

9.88 315 

26 

10.11 685 

9.90 014 

10 

37 

24 

9.78 346 

17 

1 A 

9.88 341 

26 

9A 

10.11 659 

9.90 005 

9 

10 

36 

25 

9.78 362 

iO 

9.88 367 

ZD 

10.11 633 

9.89 995 

35 

26 

9.78 379 

17 

9.88 393 

26 

10.11 607 • 

9.89 985 

10 

34 

27 

9.78 395 

16 

9.88 420 

27 

10.11 580 

9.89 976 

9 

33 

28 

9.78 412 

17 

9.88 446 

26 

10.11 554 

9.89 966 

10 

32 

29 

9.78 428 

16 

9.88 472 

26 

9A 

10.11 528 

9.89 956 

10 

g 

31 

30 

9.78 445 

17 

9.88 498 

ZO 

10.11 502 

9.89 947 


30 

31 

9.78 461 

16 

9.88 524 

26 

10.11 476 

9.89 937 

10 

29 

32 

9.78 478 

17 

9.88 550 

26 

10.11 450 

9.89 927 

10 

28 

33 

9.78 494 

16 

9.88 577 

27 

10.11 423 

9.89 918 

9 

27 

34 

9.78 510 

16 

1 7 

9.88 603 

26 

26 

10.11 397 

9.89 908 

10 

10 

26 

35 

9.78 527 

1 l 

9.88 629 


10.11 371 

9.89 898 

25 

36 

9.78 543 

16 

9.88 655 

26 

10.11 345 

9.89 888 

10 

24 

37 

9.78 560 

17 

9.88 681 

26 

10.11 319 

9.89 879 

9 

23 

38 

9.78 576 

16 

9.88 707 

26 

10.11 293 

9.89 869 

10 

22 

39 

9.78 592 

16 

1 7 

9.88 733 

26 

9A 

10.11 267 

9.89 859 

10 
i n 

21 

40 

9.78 609 


9.88 759 

ZO 

10.11 241 

9.89 849 


20 

41 

9.78 625 

16 

9.88 786 

27 

10.11 214 

9.89 840 

9 

19 

42 

9.78 642 

17 

9.88 812 

26 

10.11 188 

9.89 830 

10 

18 

43 

9.78 658 

16 

9.88 838 

26 

10.11 162 

9.89 820 

10 

17 

44 

9.78 674 

16 

1 7 

9.88 864 

26 

26 

10.11 136 

9.89 810 

10 

Q 

16 

45 

9.78 691 

-1 1 

9.88 890 

10.11 110 

9.89 801 

57 

15 

46 

9.78 707 

16 

9.88 916 

26 

10.11 084 

9.89 791 

10 

14 

47 

9.78 723 

16 

9.88 942 

26 

10.11 058 

9.89 781 

10 

13 

48 

9.78 739 

16 

9.88 968 

26 

10.11 032 

9.89 771 

10 

12 

49 

9.78 756 

17 

1 A 

9.88 994 

26 

9A 

10.11 006 

9.89 761 

10 

g 

11 

50 

9.78 772 

J.O 

9.89 020 

ZO 

10.10 980 

9.89 752 


10 

51 

9.78 788 

16 

9.89 046 

26 

10.10 954 

9.89 742 

10 

9 

52 

9.78 805 

17 

9.89 073 

27 

10.10 927 

9.89 732 

10 

8 

53 

9.78 821 

16 

9.89 099 

26 

10.10 901 

9.89 722 

10 

7 

54 

9.78 837 

16 

1 A 

9.89 125 

26 

26 

10.10 875 

9.89 712 

10 

10 

6 

55 

9.78 853 

ID 

9.89 151 

10.10 849 

9.89 702 

5 

56 

9.78 869 

16 

9.89 177 

26 

10.10 823 

9.89 693 

9 

4 

57 

9.78 886 

17 

9.89 203 

26 

10.10 797 

9.89 683 

10 

3 

58 

9.78 902 

16 

9.89 229 

26 

10.10 771 

9.89 673 

10 

2 

59 

9.78 918 

16 

16 

9,89 255 

26 

26. 

10.10 745 

9.89 663 

10 

10 

1 

60 

9.78 934 

9.89 281 

10.10 719 

9.89 653 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 


Prop. Pts. 


" 

27 

26 

17 

6 

2.7 

2.6 

1.7 

7 

3.2 

3.0 

2.0 

8 

3.6 

3.5 

2.3 

9 

4.0 

3.9 

2.6 

10 

4.5 

4.3 

2.8 

20 

9.0 

8.7 

5.7 

30 

13.5 

13.0 

8.5 

40 

18.0 

17.3 

11.3 

50 

22.5 

21.7 

14.2 


" 

16 

10 

9 

6 

1.6 

1.0 

0.9 

7 

1.9 

1.2 

1.0 

8 

2.1 

1.3 

1.2 

9 

2.4 

1.5 

1.4 

10 

2.7 

1.7 

1.5 

20 

5.3 

3.3 

3.0 

30 

8.0 

5.0 

4.5 

40 

10.7 

6.7 

6.0 

50 

13.3 

8.3 

7.5 


Prop. Pts. 


52 ( 












































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


65 


38 c 


/ 

L Sin 

d 

L Tan 

cd 

L Cot 

L Cos 

d 


0 

9.78 934 


9.89 281 


10.10 719 

9.89 653 


60 

1 

9.78 950 

16 

9.89 307 

26 

10.10 693 

9.89 643 

10 

59 

2 

9.78 967 

17 

9.89 333 

26 

10.10 667 

9.89 633 

10 

58 

3 

9.78 983 

16 

9.89 359 

26 

10.10 641 

9.89 624 

9 

57 

4 

9.78 999 

16 

16 

9.89 385 

26 

26 

10.10 615 

9.89 614 

10 

56 

5 

9.79 015 

9.89 411 

10.10 589 

9.89 604 

10 

55 

6 

9.79 031 

16 

9.89 437 

26 

10.10 563 

9.89 594 

10 

54 

7 

9.79 047 

16 

9.89 463 

26 

10.10 537 

9.89 584 

10 

53 

8 

9.79 063 

16 

9.89 489 

26 

10.10 511 

9.89 574 

10 

52 

9 

9.79 079 

16 

16 

9.89 515 

26 

26 

10.10 485 

9.89 564 

10 

51 

10 

9.79 095 

9.89 541 

10.10 459 

9.89 554 

10 

50 

11 

9.79 111 

16 

9.89 567 

26 

10.10 433 

9.89 544 

10 

49 

12 

9.79 128 

17 

9.89 593 

26 

10.10 407 

9.89 534 

10 

48 

13 

9.79 144 

16 

9.89 619 

26 

10.10 381 

9.89 524 

10 

47 

14 

9.79 160 

16 

16 

9.89 645 

26 

26 

10.10 355 

9.89 514 

10 

46 

15 

9.79 176 

9.89 671 

10.10 329 

9.89 504 

10 

45 

16 

9.79 192 

16 

9.89 697 

26 

10.10 303 

9.89 495 

9 

44 

17 

9.79 208 

16 

9.89 723 

26 

10.10 277 

9.89 485 

10 

43 

18 

9.79 224 

16 

9.89 749 

26 

10.10 251 

9.89 475 

10 

42 

19 

9.79 240 

16 

16 

9.89 775 

26 

26 

10.10 225 

9.89 465 

10 

41 

20 

9.79 256 

9.89 801 

10.10 199 

9.89 455 

10 

40 

21 

9.79 272 

16 

9.89 827 

26 

10.10 173 

9.89 445 

10 

39 

22 

9.79 288 

16 

9.89 853 

26 

10.10 147 

9.89 435 

10 

38 

23 

9.79 304 

16 

9.89 879 

26 

10.10 121 

9.89 425 

10 

37 

24 

9.79 319 

15 

16 

9.89 905 

26 

26 

10.10 095 

9.89 415 

10 

36 

25 

9.79 335 

9.89 931 

10.10 069 

9.89 405 

10 

35 

26 

9.79 351 

16 

9.89 957 

26 

10.10 043 

9.89 395 

10 

34 

27 

9.79 367 

16 

9.89 983 

26 

10.10 017 

9.89 385 

10 

33 

28 

9.79 383 

16 

9.90 009 

26 

10.09 991 

9.89 375 

10 

32 

29 

9.79 399 

16 

16 

9.90 035 

26 

26 

10.09 965 

9.89 364 

11 

31 

30 

9.79 415 

9.90 061 

10.09 939 

9.89 354 

10 

30 

31 

9.79 431 

16 

9.90 086 

25 

10.09 914 

9.89 344 

10 

29 

32 

9.79 447 

16 

9.90 112 

26 

10.09 888 

9.89 334 

10 

28 

33 

9.79 463 

16 

9.90 138 

26 

10.09 862 

9.89 324 

10 

27 

34 

9.79 478 

15 

16 

9.90 164 

26 

26 

10.09 836 

9.89 314 

10 

10 

26 

35 

9.79 494 

9.90 190 

10.09 810 

9.89 304 

25 

36 

9.79 510 

16 

9.90 216 

26 

10.09 784 

9.89 294 

10 

24 

37 

9.79 526 

16 

9.90 242 

26 

10.09 758 

9.89 284 

10 

23 

38 

9.79 542 

16 

9.90 268 

26 

10.09 732 

9.89 274 

10 

22 

39 

9.79 558 

16 

15 

9.90 294 

26 

26 

10.09 706 

9.89 264 

10 

10 

21 

40 

9.79 573 

9.90 320 

10.09 680 

9.89 254 

20 

41 

9.79 589 

16 

9.90 346 

26 

10.09 654 

9.89 244 

10 

19 

42 

9.79 605 

16 

9.90 371 

25 

10.09 629 

9.89 233 

11 

18 

43 

9.79 621 

16 

9.90 397 

26 

10.09 603 

9.89 223 

10 

17 

44 

9.79 636 

15 

1 A 

9.90 423 

26 

26 

10.09 577 

9.89 213 

10 

1 A 

16 

45 

9.79 652 

10 

9.90 449 

10.09 551 

9.89 203 

1U 

15 

46 

9.79 668 

16 

9.90 475 

26 

10.09 525 

9.89 193 

10 

14 

47 

9.79 684 

16 

9.90 501 

26 

10.09 499 

9.89 183 

10 

13 

48 

9.79 699 

15 

9.90 527 

26 

10.09 473 

9.89 173 

10 

12 

49 

9.79 715 

16 

1 A 

9.90 553 

26 

oi: 

10.09 447 

9.89 162 

11 

10 

11 

50 

9.79 731 

1 O 

9.90 578 


10.09 422 

9.89 152 

10 

51 

9.79 746 

15 

9.90 604 

26 

10.09 396 

9.89 142 

10 

9 

52 

9.79 762 

16 

9.90 630 

26 

10.09 370 

9.89 132 

10 

8 

53 

9.79 778 

16 

9.90 656 

26 

10.09 344 

9.89 122 

10 

7 

54 

9.79 793 

15 

1 A 

9.90 682 

26 

10.09 318 

9.89 112 

10 

11 

6 

55 

9.79 809 

ID 

9.90 708 

40 

10.09 292 

9.89 101 

5 

56 

9.79 825 

16 

9.90 734 

26 

10.09 266 

9.89 091 

10 

4 

57 

9.79 840 

15 

9.90 759 

25 

10.09 241 

9.89 081 

10 

3 

58 

9.79 856 

16 

9.90 785 

26 

10.09 215 

9.89 071 

10 

2 

59 

9.79 872 

16 

9.90 811 

26 

10.09 189 

9.89 060 

11 

1 








10 


60 

9.79 887 

lo 

9.90 837 


10.09 163 

9.89 050 

0 

| L Cos 

d 

L Cot 

cd 

L Tan 

L Sin 

d 

' 


Prop. Pts. 


26 

2.6 

3.0 

3.5 

3.9 

4.3 

8.7 

13.0 

17.3 

21.7 


25 

2.5 

2.9 

3.3 

3.8 

4.2 

8.3 
12.5 

16.7 

20.8 


16 

1.6 

1.9 

2.1 

2.4 

2.7 

5.3 

8.0 

10.7 

13.3 


15 

1.5 
1.8 
2.0 
2.2 

2.5 
5.0 

7.5 
10.0 
12.5 


Prop. Pts. 


51 ' 











































































66 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


39 c 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

0 

9.79 887 


9.90 837 


10.09 163 

9.89 050 

1 

9.79 903 

16 

9.90 863 

26 

10.09 137 

9.89 040 

2 

9.79 918 

15 

9.90 889 

26 

10.09 111 

9.89 030 

3 

9.79 934 

16 

9.90 914 

25 

10.09 086 

9.89 020 

4 

9.79 950 

16 

9.90 940 

26 

26 

10.09 060 

9.89 009 

5 

9.79 965 


9.90 966 

10.09 034 

9.88 999 

6 

9.79 981 

16 

9.90 992 

26 

10.09 008 

9.88 989 

7 

9.79 996 

15 

9.91 018 

26 

10.08 982 

9.88 978 

8 

9.80 012 

16 

9.91 043 

25 

10.08 957 

9.88 968 

9 

9.80 027 

15 

9.91 069 

26 

Oft 

10.08 931 

9.88 958 

10 

9.80 043 


9.91 095 


10.08 905 

9.88 948 

11 

9.80 058 

15 

9.91 121 

26 

10.08 879 

9.88 937 

12 

9.80 074 

16 

9.91 147 

26 

10.08 853 

9.88 927 

13 

9.80 089 

15 

9.91 172 

25 

10.08 828 

9.88 917 

14 

9.80 105 

16 

9.91 198 

26 

26 

10.08 802 

9.88 906 

15 

9.80 120 


9.91 224 

10.08 776 

9.88 896 

16 

9.80 136 

16 

9.91 250 

26 

10.08 750 

9.88 886 

17 

9.80 151 

15 

9.91 276 

26 

10.08 724 

9.88 875 

18 

9.80 166 

15 

9.91 301 

25 

10.08 699 

9.88 865 

19 

9.80 182 

16 

9.91 327 

26 

Oft 

10.08 673 

9.88 855 

20 

9.80 197 


9.91 353 

ZO 

10.08 647 

9.88 844 

21 

9.80 213 

16 

9.91 379 

26 

10.08 621 

9.88 834 

22 

9.80 228 

15 

9.91 404 

25 

10.08 596 

9.88 824 

23 

9.80 244 

16 

9.91 430 

26 

10.08 570 

9.88 813 

24 

9.80 259 

15 

9.91 456 

26 

Oft 

10.08 544 

9.88 803 

25 

9.80 274 

id 

9.91 482 

ZD 

10.08 518 

9.88 793 

26 

9.80 290 

16 

9.91 507 

25 

10.08 493 

9.88 782 

27 

9.80 305 

15 

9.91 533 

26 

10.08 467 

9.88 772 

28 

9.80 320 

15 

9.91 559 

26 

10.08 441 

9.88 761 

29 

9.80 336 

16 

1 p; 

9.91 585 

26 

10.08 415 

9.88 751 

30 

9.80 351 

ID 

9.91 610 

zo 

10.08 390 

9.88 741 

31 

9.80 366 

15 

9.91 636 

26 

10.08 364 

9.88 730 

32 

9.80 382 

16 

9.91 662 

26 

10.08 338 

9.88 720 

33 

9.80 397 

15 

9.91 688 

26 

10.08 312 

9.88 709 

34 

9.80 412 

15 

16 

9.91 713 

25 

26 

10.08 287 

9.88 699 

35 

9.80 428 


9.91 739 

10.08 261 

9.88 688 

36 

9.80 443 

15 

9.91 765 

26 

10.08 235 

9.88 678 

37 

9.80 458 

15 

9.91 791 

26 

10.08 209 

9.88 668 

38 

9.80 473 

15 

9.91 816 

25 

10.08 184 

9.88 657 

39 

9.80 489 

16 
i p; 

9.91 842 

26 

9ft 

10.08 158 

9.88 647 

40 

9.80 504 

ID 

9.91 868 

ZD 

10.08 132 

9.88 636 

41 

9.80 519 

15 

9.91 893 

25 

10.08 107 

9.88 626 

42 

9.80 534 

15 

9.91 919 

26 

10.08 081 

9.88 615 

43 

9.80 550 

16 

9.91 945 

26 

10.08 055 

9.88 605 

44 

9.80 565 

15 

9.91 971 

26 

25 

10.08 029 

9.88 594 

45 

9.80 580 

ID 

9.91 996 

10.08 004 

9.88 584 

46 

9.80 595 

15 

9.92 022 

26 

10.07 978 

9.88 573 

47 

9.80 610 

15 

9.92 048 

26 

10.07 952 

9.88 563 

48 

9.80 625 

15 

9.92 073 

25 

10.07 927 

9.88 552 

49 

9.80 641 

16 

15 

9.92 099 

26 

9ft 

10.07 901 

9.88 542 

50 

9.80 656 

9.92 125 

ZD 

10.07 875 

9.88 531 

51 

9.80 671 

15 

9.92 150 

25 

10.07 850 

9.88 521 

52 

9.80 686 

15 

9.92 176 

26 

10.07 824 

9.88 510 

53 

9.80 701 

15 

9.92 202 

26 

10.07 798 

9.88 499 

54 

9.80 716 

15 

15 

9.92 227 

25 

9ft 

10.07 773 

9.88 489 

55 

9.80 731 

9.92 253 

ZD 

10.07 747 

9.88 478 

56 

9.80 746 

15 

9.92 279 

26 

10.07 721 

9.88 468 

57 

9.80 762 

16 

9.92 304 

25 

10.07 696 

9.88 457 

58 

9.80 777 

15 

9.92 330 

26 

10.07 670 

9.88 447 

59 

9.80 792 

15 

15 

9.92 356 

26 

9 ^ 

10.07 644 

9.88 436 

60 

9.80 807 


9.92 381 

ZO 

10.07 619 

9.88 425 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 


Prop. Pts. 


26 

2.6 

3.0 

3.5 

3.9 

4.3 

8.7 

13.0 

17.3 

21.7 


25 

2.5 

2.9 

3.3 

3.8 

4.2 

8.3 
12.5 

16.7 

20.8 


16 

1.6 

1.9 

2.1 

2.4 

2.7 

5.3 

8.0 

10.7 

13.3 


15 

1.5 
1.8 
2.0 
2.2 

2.5 
5.0 

7.5 
10.0 
12.5 


Prop. Pts. 


50 ‘ 













































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


67 


40° 


' 

L Sin 

d 

L Tan 

c d 

L Cot j 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.80 807 
9.80 822 
9.80 837 
9.80 852 
9.80 867 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

15 

14 

15 
15 
15 

15 

15 

15 

15 

14 

15 
15 
15 
15 

14 

15 
15 
15 
15 

14 

15 
15 

14 

15 
15 

15 

14 

15 
15 

14 

15 
15 

14 

15 
15 

14 

15 

14 

15 
15 

14 

15 

14 

15 
14 

9.92 381 
9.92 407 
9.92 433 
9.92 458 
9.92 484 

26 

26 

25 

26 
26 

25 

26 
26 

25 

26 

25 

26 
26 

25 

26 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 
26 

25 

26 

25 

26 

25 

26 
26 

25 

26 

25 

26 

25 

26 

25 

26 
26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 
25 

10.07 619 
10.07 593 
10.07 567 
10.07 542 
10.07 516 

9.88 425 
9.88 415 
9.88 404 
9.88 394 
9.88 383 

10 

11 

10 

11 

11 

10 

11 

11 

10 

11 

11 

10 

11 

11 

10 

11 

11 

10 

11 

11 

11 

10 

11 

11 

11 

10 

11 

11 

11 

10 

11 

11 

11 

11 

10 

11 

11 

11 

11 

11 

11 

10 

11 

11 

11 

11 

11 

11 

11 

11 

10 

11 

11 

11 

11 

11 

11 

11 

11 

11 

60 

59 

58 

57 

56 

" 26 25 

6 2.6 2.5 

7 3.0 2.9 

8 3.5 3.3 

9 3.9 3.8 

10 4.3 4.2 

20 8.7 8.3 

30 13.0 12.5 

40 17.3 16.7 

50 21.7 20.8 

" 15 14 

6 1.5 1.4 

7 1.8 1.6 

8 2.0 1.9 

9 2.2 2.1 

10 2.5 2.3 

20 5.0 4.7 

30 7.5 7.0 

40 10.0 9.3 

50 12.5 11.7 

" 11 10 

6 1.1 1.0 

7 1.3 1.2 

8 1.5 1.3 

9 1.6 1.5 

10 1.8 1.7 

20 3.7 3.3 

30 5.5 5.0 

40 7.3 6.7 

50 9.2 8.3 

5 

6 

7 

8 

9 

9.80 882 
9.80 897 
9.80 912 
9.80 927 
9.80 942 

9.92 510 
9.92 535 
9.92 561 
9.92 587 
9.92 612 

10.07 490 
10.07 465 
10.07 439 
10.07 413 
10.07 388 

9.88 372 
9.88 362 
9.88 351 
9.88 340 
9.88 330 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.80 957 
9.80 972 

9.80 987 

9.81 002 
9.81 017 

9.92 638 
9.92 663 
9.92 689 
9.92 715 
9.92 740 

10.07 362 
10.07 337 
10.07 311 
10.07 285 
10.07 260 

9.88 319 
9.88 308 
9.88 298 
9.88 287 
9.88 276 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.81 032 
9.81 047 
9.81 061 
9.81 076 
9.81 091 

9.92 766 
9.92 792 
9.92 817 
9.92 843 
9.92 868 

10.07 234 
10.07 208 
10.07 183 
10.07 157 
10.07 132 

9.88 266 
9.88 255 
9.88 244 
9.88 234 
9.88 223 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.81 106 
9.81 121 
9.81 136 
9.81 151 
9.81 166 

9.92 894 
9.92 920 
9.92 945 
9.92 971 
9.92 996 

10.07 106 
10.07 080 
10.07 055 
10.07 029 
10.07 004 

9.88 212 
9.88 201 
9.88 191 
9.88 180 
9.88 169 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.81 180 
9.81 195 
9.81 210 
9.81 225 
9.81 240 

9.93 022 
9.93 048 
9.93 073 
9.93 099 
9.93 124 

10.06 978 
10.06 952 
10.06 927 
10.06 901 
10.06 876 

9.88 158 
9.88 148 
9.88 137 
9.88 126 
9.88 115 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.81 254 
9.81 269 
9.81 284 
9.81 299 
9.81 314 

9.93 150 
9.93 175 
9.93 201 
9.93 227 
9.93 252 

10.06 850 
10.06 825 
10.06 799 
10.06 773 
10.06 748 

9.88 105 
9.88 094 
9.88 083 
9.88 072 
9.88 061 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.81 328 
9.81 343 
9.81 358 
9.81 372 
9.81 387 

9.93 278 
9.93 303 
9.93 329 
9.93 354 
9.93 380 

10.06 722 
10.06 697 
10.06 671 
10.06 646 
10.06 620 

9.88 051 
9.88 040 
9.88 029 
9.88 018 
9.88 007 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.81 402 
9.81 417 
9.81 431 
9.81 446 
9.81 461 

9.93 406 
9.93 431 
9.93 457 
9.93 482 
9.93 508 

10.06 594 
10.06 569 
10.06 543 
10.06 518 
10.06 492 

9.87 996 
9.87 985 
9.87 975 
9.87 964 
9.87 953 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.81 475 
9.81 490 
9.81 505 
9.81 519 
9.81 534 

9.93 533 
9.93 559 
9.93 584 
9.93 610 
9.93 636 

10.06 467 
10.06 441 
10.06 416 
10.06 390 
10.06 364 

9.87 942 
9.87 931 
9.87 920 
9.87 909 
9.87 898 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.81 549 
9.81 563 
9.81 578 
9.81 592 
9.81 607 

9.93 661 
9.93 687 
9.93 712 
9.93 738 
9.93 763 

10.06 339 
10.06 313 
10.06 288 
10.06 262 
10.06 237 

9.87 887 
9.87 877 
9.87 866 
9.87 855 
9.87 844 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.81 622 
9.81 636 
9.81 651 
9.81 665 
9.81 680 

9.93 789 
9.93 814 
9.93 840 
9.93 865 
9.93 891 

10.06 211 
10.06 186 
10.06 160 
10.06 135 
10.06 109 

9.87 833 
9.87 822 
9.87 811 
9.87 800 
9.87 789 

5 

4 

3 

2 

1 

60 

9.81 694 

9.93 916 

10.06 084 

9.87 778 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


49 ' 









































































68 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


41° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

o 

9.81 694 


9.93 916 

26 

10.06 

084 

9.87 

778 

1 

9.81 709 

15 

9.93 942 

10.06 

058 

9.87 

767 

2 

9.81 723 

14 

9.93 967 

25 

10.06 

033 

9.87 

756 

3 

9.81 738 

15 

9.93 993 

26 

10.06 

007 

9.87 

745 

4 

9.81 752 

14 

9.94 018 

25 

10.05 

982 

9.87 

734 

5 

9.81 767 


9.94 044 

25 

10.05 

956 

9.87 

723 

6 

9.81 781 

14 

9.94 069 

10.05 

931 

9.87 

712 

7 

9.81 796 

15 

9.94 095 

26 

10.05 

905 

9.87 

701 

8 

9.81 810 

14 

9.94 120 

25 

10.05 

880 

9.87 

690 

9 

9.81 825 

15 

14 

9.94 146 

26 

25 

10.05 

854 

9.87 

679 

10 

9.81 839 


9.94 171 

26 

10.05 

829 

9.87 

668 

11 

9.81 854 

15 

9.94 197 

10.05 

803 

9.87 

657 

12 

9.81 868 

14 

9.94 222 

25 

10.05 

778 

9.87 

646 

13 

9.81 882 

14 

9.94 248 

26 

10.05 

752 

9.87 

635 

14 

9.81 897 

15 

9.94 273 

25 

10.05 

727 

9.87 

624 

15 

9.81 911 


9.94 299 

26 

10.05 

701 

9.87 

613 

16 

9.81 926 

15 

9.94 324 

25 

10.05 

676 

9.87 

601 

17 

9.81 940 

14 

9.94 350 

26 

10.05 

650 

9.87 

590 

18 

9.81 955 

15 

9.94 375 

25 

10.05 

625 

9.87 

579 

19 

9.81 969 

14 

9.94 401 

26 

25 

10.05 

599 

9.87 

568 

20 

9.81 983 

14 

9.94 426 


10.05 

574 

9.87 

557 

21 

9.81 998 

15 

9.94 452 

26 

10.05 

548 

9.87 

546 

22 

9.82 012 

14 

9.94 477 

25 

10.05 

523 

9.87 

535 

23 

9.82 026 

14 

9.94 503 

26 

10.05 

497 

9.87 

524 

24 

9.82 041 

15 

9.94 528 

25 

10.05 

472 

9.87 

513 

25 

9.82 055 

14 

9.94 554 

25 

10.05 

446 

9.87 

501 

26 

9.82 069 

14 

9.94 579 

10.05 

421 

9.87 

490 

27 

9.82 084 

15 

9.94 604 

25 

10.05 

396 

9.87 

479 

28 

9.82 098 

14 

9.94 630 

26 

10.05 

370 

9.87 

468 

29 

9.82 112 

14 

9.94 655 

25 

10.05 

345 

9.87 

457 

30 

9.82 126 

14 

9.94 681 

25 

10.05 

319 

9.87 

446 

31 

9.82 141 

15 

9.94 706 

10.05 

294 

9.87 

434 

32 

9.82 155 

14 

9.94 732 

26 

10.05 

268 

9.87 

423 

33 

9.82 169 

14 

9.94 757 

25 

10.05 

243 

9.87 

412 

34 

9.82 184 

15 

9.94 783 

26 

25 

10.05 

217 

9.87 

401 

35 

9.82 198 

14 

9.94 808 


10.05 

192 

9.87 

390 

36 

9.82 212 

14 

9.94 834 

26 

10.05 

166 

9.87 

378 

37 

9.82 226 

14 

9.94 859 

25 

10.05 

141 

9.87 

367 

38 

9.82 240 

14 

9.94 884 

25 

10.05 

116 

9.87 

356 

39 

9.82 255 

15 

9.94 910 

26 

10.05 

090 

9.87 

345 

40 

9.82 269 

14 

9.94 935 

26 

10.05 

065 

9.87 

334 

41 

9.82 283 

14 

9.94 961 

10.05 

039 

9.87 

322 

42 

9.82 297 

14 

9.94 986 

25 

10.05 

014 

9.87 

311 

43 

9.82 311 

14 

9.95 012 

26 

10.04 

988 

9.87 

300 

44 

9.82 326 

15 

9.95 037 

25 

10.04 

963 

9.87 

288 

45 

9.82 340 

14 

9.95 062 

26 

10.04 

938 

9.87 

277 

46 

9.82 354 

14 

9.95 088 

10.04 

912 

9.87 

266 

47 

9.82 368 

14 

9.95 113 

25 

10.04 

887 

9.87 

255 

48 

9.82 382 

14 

9.95 139 

26 

10.04 

861 

9.87 

243 

49 

9.82 396 

14 

9.95 164 

25 

OR 

10.04 

836 

9.87 

232 

50 

9.82 410 

14 

9.95 190 

25 

10.04 

810 

9.87 

221 

51 

9.82 424 

14 

9.95 215 

10.04 

785 

9.87 

209 

52 

9.82 439 

15 

9.95 240 

25 

10.04 

760 

9.87 

198 

53 

9.82 453 

14 

9.95 266 

26 

10.04 

734 

9.87 

187 

54 

9.82 467 

14 

9.95 291 

25 

10.04 

709 

9.87 

175 

55 

9.82 481 

14 

9.95 317 

25 

10.04 

683 

9.87 

164 

56 

9.82 495 

14 

9.95 342 

10.04 

658 

9.87 

153 

57 

9.82 509 

14 

9.95 368 

26 

10.04 

632 

9.87 

141 

58 

9.82 523 

14 

9.95 393 

25 

10.04 

607 

9.87 

130 

59 

9.82 537 

14 

14 

9.95 418 

25 

10.04 

582 

9.87 

119 

60 

9.82 551 

9.95 444 


10.04 

556 

9.87 

107 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 


Prop. Pts. 


2.5 

2.9 

3.3 

3.8 

4.2 

8.3 
12.5 

16.7 

20.8 


14 

1.4 

1.6 

1.9 

2.1 

2.3 
4.7 
7.0 

9.3 
11.7 


Prop. Pts. 


48 ' 













































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


69 


42° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

9.82 551 


9.95 444 


10.04 556 

9.87 107 


60 




1 

9.82 565 

14 

9.95 469 

25 

10.04 531 

9.87 096 

11 

59 




2 

9.82 579 

14 

9.95 495 

26 

10.04 505 

9.87 085 

11 

58 




3 

9.82 593 

14 

9.95 520 

25 

10.04 480 

9.87 073 

12 

57 




4 

9.82 607 

14 

14 

9.95 545 

25 

26 

10.04 455 

9.87 062 

11 

12 

56 




5 

9.82 621 

9.95 571 

10.04 429 

9.87 050 

55 




6 

9.82 635 

14 

9.95 596 

25 

10.04 404 

9.87 039 

11 

54 




7 

9.82 649 

14 

9.95 622 

26 

10.04 378 

9.87 028 

11 

53 

„ 

26 

25 

8 

9.82 663 

14 

9.95 647 

25 

10.04 353 

9.87 016 

12 

52 


9 

9.82 677 

14 

14 

9.95 672 

25 

26 

10.04 328 

9.87 005 

11 

12 

51 

6 

2.6 

2.5 

10 

9.82 691 

9.95 698 

10.04 302 

9.86 993 

50 

7 

g 

3.0 

o K 

2.9 

5 9 

11 

9.82 705 

14 

9.95 723 

25 

10.04 277 

9.86 982 

11 

49 

9 

0.0 

3.9 

0.0 

3.8 

12 

9.82 719 

14 

9.95 748 

25 

10.04 252 

9.86 970 

12 

48 

10 

4.3 

4.2 

13 

9.82 733 

14 

9.95 774 

26 

10.04 226 

9.86 959 

11 

47 

20 

8.7 

8.3 

14 

9.82 747 

14 

14 

9.95 799 

25 

26 

10 04 201 

9.86 947 

12 

11 

46 

30 

40 

13.0 
17 3 

12.5 

16 7 

15 

9.82 761 

9.95 825 

10.04 175 

9.86 936 

45 

50 

21.7 

20^8 

16 

9.82 775 

14 

9.95 850 

25 

10.04 150 

9.86 924 

12 

44 




17 

9.82 788 

13 

9.95 875 

25 

10.04 125 

9.86 913 

11 

43 




18 

9.82 802 

14 

9.95 901 

26 

10.04 099 

9.86 902 

11 

42 




19 

9.82 816 

14 

14 

9.95 926 

25 

26 

10.04 074 

9.86 890 

12 

11 

41 




20 

9.82 830 

9.95 952 

10.04 048 

9.86 879 

40 




21 

9.82 844 

14 

9.95 977 

25 

10.04 023 

9.86 867 

12 

39 




22 

9.82 858 

14 

9.96 002 

25 

10.03 998 

9.86 855 

12 

38 




23 

9.82 872 

14 

9.96 028 

26 

10.03 972 

9.86 844 

11 

37 




24 

9.82 885 

13 

14 

9.96 053 

25 

25 

10.03 947 

9.86 832 

12 

11 

36 




25 

9.82 899 

9.96 078 

10.03 922 

9.86 821 

35 




26 

9.82 913 

14 

9.96 104 

26 

10.03 896 

9.86 809 

12 

34 

// 

14 

13 

27 

9.82 927 

14 

9.96 129 

25 

10.03 871 

9.86 798 

11 

33 




28 

9.82 941 

14 

9.96 155 

26 

10.03 845 

9.86 786 

12 

32 

6 

1.4 

1.3 

29 

9.82 955 

14 

1 1 

9.96 180 

25 

OK 

10.03 820 

9.86 775 

11 

12 

31 

7 

8 

1.6 

1.9 

1.5 

1.7 

30 

9.82 968 

l o 

9.96 205 

AO 

10.03 795 

9.86 763 

30 

9 

2.1 

2.0 

31 

9.82 982 

14 

9.96 231 

26 

10.03 769 

9.86 752 

11 

29 

10 

2.3 

2.2 

32 

9.82 996 

14 

9.96 256 

25 

10.03 744 

9.86 740 

12 

28 

20 

QA 

4.7 

4.3 

33 

9.83 010 

14 

9.96 281 

25 

10.03 719 

9.86 728 

12 

27 

oU 

40 

/ .1) 
9 3 

8 7 

34 

9.83 023 

13 

1 A 

9.96 307 

26 

OK 

10.03 693 

9.86 717 

11 

12 

26 

50 

11.7 

10.8 

35 

9.83 037 

Art 

9.96 332 

AO 

10.03 668 

9.86 705 

25 




36 

9.83 051 

14 

9.96 357 

25 

10.03 643 

9.86 694 

11 

24 




37 

9.83 065 

14 

9.96 383 

26 

10.03 617 

9.86 682 

12 

23 




38 

9.83 078 

13 

9.96 408 

25 

10.03 592 

9.86 670 

12 

22 




39 

9.83 092 

14 

9.96 433 

25 

OR 

10.03 567 

9.86 659 

11 

12 

21 




40 

9.83 106 

14 

9.96 459 

AO 

10.03 541 

9.86 647 

20 




41 

9.83 120 

14 

9.96 484 

25 

10.03 516 

9.86 635 

12 

19 




42 

9.83 133 

13 

9.96 510 

26 

10.03 490 

9.86 624 

11 

18 




43 

9.83 147 

14 

9.96 535 

25 

10.03 465 

9.86 612 

12 

17 




44 

9.83 161 

14 

1 Q 

9.96 560 

25 

OR 

10.03 440 

9.86 600 

12 

11 

16 




45 

9.83 174 

1 6 

9.96 586 

AO 

10.03 414 

9.86 589 

15 

" 

12 

11 

46 

9.83 188 

14 

9.96 611 

25 

10.03 389 

9.86 577 

12 

14 




47 

9.83 202 

14 

9.96 636 

25 

10.03 364 

9.86 565 

12 

13 

6 

7 

1.2 
1 4 

1.1 

1.3 

48 

9.83 215 

13 

9.96 662 

26 

10.03 338 

9.86 554 

11 

12 

8 

1*6 

1.5 

49 

9.83 229 

14 

9.96 687 

25 

(>C 

10.03 313 

9.86 542 

12 

12 

11 

9 

1.8 

1.6 

50 

9.83 242 

13 

9.96 712 

AO 

10.03 288 

9.86 530 

10 

10 

20 

2.0 
a n 

1.8 

3 7 

51 

9.83 256 

14 

9.96 738 

26 

10.03 262 

9.86 518 

12 

9 

30 

rt-KJ 

6.0 

5.5 

52 

9.83 270 

14 

9.96 763 

25 

10.03 237 

9.86 507 

11 

8 

40 

8.0 

7.3 

53 

9.83 283 

13 

9.96 788 

25 

10.03 212 

9.86 495 

12 

7 

50 

10.0 

9.2 

54 

9.83 297 

14 

9.96 814 

26 

OK 

10.03 186 

9.86 483 

12 

11 

6 




~55~ 

9.83 310 

13 

9.96 839 

AO 

10.03 161 

9.86 472 

5 




56 

9.83 324 

14 

9.96 864 

25 

10.03 136 

9.86 460 

12 

4 




57 

9.83 338 

14 

9.96 890 

26 

10.03 110 

9.86 448 

12 

3 




58 

9.83 351 

13 

9.96 915 

25 

10.03 085 

9.86 436 

12 

2 




59 

9.83 365 

14 

. 9.96 940 

25 

OR 

10.03 060 

9.86 425 

11 

12 

1 




60 

9.83 378 

13 

9.96 966 

AO 

10.03 034 

9.86 413 

0 





L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


47 * 










































































70 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


VI 


43 ° 


' 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop . Pts . 

0 

9.83 

378 

14 

9.96 

966 

25 

10.03 

034 

9.86 

413 

12 

60 




1 

9.83 

392 

9.96 

991 

10.03 

009 

9.86 

401 

59 




2 

9.83 

405 

13 

9.97 

016 

25 

10.02 

984 

9.86 

389 

12 

58 




3 

9.83 

419 

14 

9.97 

042 

26 

10.02 

958 

9.86 

377 

12 

57 




4 

9.83 

432 

13 

9.97 

067 

25 

10.02 

933 

9.86 

366 

11 

12 

56 




5 

9.83 

446 


9.97 

092 


10.02 

908 

9.86 

354 

55 




6 

9.83 

459 

13 

9.97 

118 

26 

10.02 

882 

9.86 

342 

12 

54 




7 

9.83 

473 

14 

9.97 

143 

25 

10.02 

857 

9.86 

330 

12 

53 

// 

26 

25 

8 

9.83 

486 

13 

9.97 

168 

25 

10.02 

832 

9.86 

318 

12 

52 



9 

9.83 

500 

14 

9.97 

193 

2£> 

10.02 

807 

9.86 

306 

12 

11 

51 

6 

7 

2.6 

2.5 

2 9 

10 

9.83 

513 


9.97 

219 


10.02 

781 

9.86 

295 


50 

8 

o.U 

3.5 

3.3 

11 

9.83 

527 

14 

9.97 

244 

25 

10.02 

756 

9.86 

283 

12 

49 

9 

3.9 

3.8 

12 

9.83 

540 

13 

9.97 

269 

25 

10.02 

731 

9.86 

271 

12 

48 

10 

4.3 

4.2 

13 

9.83 

554 

14 

9.97 

295 

26 

10.02 

705 

9.86 

259 

12 

47 

20 

8.7 

8.3 

14 

9.83 

567 

13 

9.97 

320 

25 

10.02 

680 

9.86 

247 

12 

12 

46 

30 

40 

13.0 

17.3 

12.5 

16.7 

15 

9.83 

581 


9.97 

345 


10.02 

655 

9.86 

235 

45 

50 

21.7 

20.8 

16 

9.83 

594 

13 

9.97 

371 

26 

10.02 

629 

9.86 

223 

12 

44 




17 

9.83 

608 

14 

9.97 

396 

25 

10.02 

604 

9.86 

211 

12 

43 




18 

9.83 

621 

13 

9.97 

421 

25 

10.02 

579 

9.86 

200 

11 

42 




19 

9.83 

634 

13 

9.97 

447 

26 

10.02 

553 

9.86 

188 

12 

12 

41 




20 

9.83 

648 


9.97 

472 


10.02 

528 

9.86 

176 

40 




21 

9.83 

661 

13 

9.97 

497 

25 

10.02 

503 

9.86 

164 

12 

39 




22 

9.83 

674 

13 

9.97 

523 

26 

10.02 

477 

9.86 

152 

12 

38 




23 

9.83 

688 

14 

9.97 

548 

25 

10.02 

452 

9.86 

140 

12 

37 




24 

9.83 

701 

13 

1 A 

9.97 

573 

25 

OK 

10.02 

427 

9.86 

128 

12 

12 

36 




25 

9.83 

715 


9.97 

598 

40 

10.02 

402 

9.86 

116 

35 




26 

9.83 

728 

13 

9.97 

624 

26 

10.02 

376 

9.86 

104 

12 

34 


14 

13 

27 

9.83 

741 

13 

9.97 

649 

25 

10.02 

351 

9.86 

092 

12 

33 

6 

1.4 

1.3 

28 

9.83 

755 

14 

9.97 

674 

25 

10.02 

326 

9.86 

080 

12 

32 

29 

9.83 

768 

13 

1 O 

9.97 

700 

26 

O K 

10.02 

300 

9.86 

068 

12 

12 

31 

7 

8 

1.6 

1.9 

1.5 

1.7 

30 

9.83 

781 

16 

9.97 

725 


10.02 

275 

9.86 

056 

30 

9 

2.1 

2.0 

31 

9.83 

795 

14 

9.97 

750 

25 

10.02 

250 

9.86 

044 

12 

29 

10 

2.3 

2.2 

32 

9.83 

808 

13 

9.97 

776 

26 

10.02 

224 

9.86 

032 

12 

28 

20 

30 

4.7 
7 0 

4.3 

6 5 

33 

9.83 

821 

13 

9.97 

801 

25 

10.02 

199 

9.86 

020 

12 

27 

40 

9^3 

8.7 

34 

9.83 

834 

13 

1 A 

9.97 

826 

25 

OK 

10.02 

174 

9.86 

008 

12 

12 

26 

50 

11.7 

10.8 

35 

9.83 

848 

l^t 

9.97 

851 

40 

10.02 

149 

9.85 

996 

25 




36 

9.83 

861 

13 

9.97 

877 

26 

10.02 

123 

9.85 

984 

12 

24 




37 

9.83 

874 

13 

9.97 

902 

25 

10.02 

098 

9.85 

972 

12 

23 




38 

9.83 

887 

13 

9.97 

927 

25 

10.02 

073 

9.85 

960 

12 

22 




39 

9.83 

901 

14 

9.97 

953 

26 

O K 

10.02 

047 

9.85 

948 

12 

12 

21 




40 

9.83 

914 

13 

9.97 

978 

40 

10.02 

022 

9.85 

936 

20 




41 

9.83 

927 

13 

9.98 

003 

25 

10.01 

997 

9.85 

924 

12 

19 




42 

9.83 

940 

13 

9.98 

029 

26 

10.01 

971 

9.85 

912 

12 

18 




43 

9.83 

954 

14 

9.98 

054 

25 

10.01 

946 

9.85 

900 

12 

17 




44 

9.83 

967 

13 

9.98 

079 

25 

O K 

10.01 

921 

9.85 

888 

12 

12 

16 




45 

9.83 

980 

13 

9.98 

104 

40 

10.01 

896 

9.85 

876 

12 

15 

" 

12 

11 

46 

9.83 

993 

13 

9.98 

130 

26 

10.01 

870 

9.85 

864 

14 

6 

7 

1.2 

1 A 


47 

9.84 

006 

13 

9.98 

155 

25 

10.01 

845 

9.85 

851 

13 

13 

1.1 

i 

48 

9.84 

020 

14 

9.98 

180 

25 

10.01 

820 

9.85 

839 

12 

12 

i 

8 

1.6 

1.0 

1.5 

49 

9.84 

033 

13 

9.98 

206 

26 

OK 

10.01 

794 

9.85 

827 

12 

12 

11 

9 

1.8 

1.6 

50 

9.84 

046 

io 

9.98 

231 

40 

10.01 

769 

9.85 

815 

12 

10 

10 

on 

2.0 
a n 

1.8 

Q 7 

51 

9.84 

059 

13 

9.98 

256 

25 

10.01 

744 

9.85 

803 

9 

4U 

30 

O.U 

6.0 

0 .1 

5.5 

52 

9.84 

072 

13 

9.98 

281 

25 

10.01 

719 

9.85 

791 

12 

8 

40 

8.0 

7.3 

53 

9.84 

085 

13 

9.98 

307 

26 

10.01 

693 

9.85 

779 

12 

7 

50 

10.0 

9.2 

54 

9.84 

098 

13 

14 

9.98 

332 

25 

OK 

10.01 

668 

9.85 

766 

13 

12 

6 




55 

9.84 

112 

9.98 

357 

40 

26 

10.01 

643 

9.85 

754 

12 

5 




56 

9.84 

125 

13 

9.98 

383 

10.01 

617 

9.85 

742 

4 




57 

9.84 

138 

13 

9.98 

408 

25 

10.01 

592 

9.85 

730 

12 

3 




58 

9.84 

151 

13 

9.98 

433 

25 

10.01 

567 

9.85 

718 

12 

2 




59 

9.84 

164 

13 

9.98 

458 

25 

OR 

10.01 

542 

9.85 

706 

12 

13 

1 




60 

9.84 

177 

13 

9.98 

00 

40 

10.01 

516 

9.85 

693 


0 





L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop . Pts . 


46 ‘ 





































































VI 


FIVE-PLACE LOGARITHMS OF FUNCTIONS 


71 


44 ° 


* 

L Sin 

d 

L Tan 

c d 

L Cot 

L Cos 

d 


Prop. Pts. 

0 

1 

2 

3 

4 

9.84 177 
9.84 190 
9.84 203 
9.84 216 
9.84 229 

13 

13 

13 

13 

13 

13 

14 
13 
13 
13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

12 

13 

13 

12 

13 

13 

9.98 484 
9.98 509 
9.98 534 
9.98 560 
9.98 585 

25 

25 

26 
25 
25 

25 

26 
25 

25 

26 

25 

25 

25 

26 
25 

25 

25 

26 
25 

25 

26 
25 
25 

25 

26 

25 

25 

25 

26 
25 

25 

26 
25 
25 

25 

26 
25 
25 

25 

26 

25 

25 

25 

26 
25 

25 

26 
25 
25 

25 

26 
25 
25 

25 

26 

25 

25 

25 

26 
25 

10.01 516 
10.01 491 
10.01 466 
10.01 440 
10.01 415 

9.85 693 
9.85 681 
9.85 669 
9.85 657 
9.85 645 

12 

12 

12 

12 

13 

12 

12 

12 

13 

12 

12 

12 

13 

12 

12 

13 

12 

12 

13 

12 

12 

13 

12 

12 

13 

12 

13 

12 

12 

13 

12 

13 

12 

13 

12 

12 

13 

12 

13 

12 

13 

12 

13 

12 

13 

12 

13 

12 

13 

13 

12 

13 

12 

13 

12 

13 

13 

12 

13 

12 

60 

59 

58 

57 

56 

" 26 25 14 

6 2.6 2.5 1.4 

7 3.0 2.9 1.6 

8 3.5 3.3 1.9 

9 3.9 3.8 2.1 

10 4.3 4.2 2.3 

20 8.7 8.3 4.7 

30 13.0 12.5 7.0 

40 17.3 16.7 9.3 

50 21.7 20.8 11.7 

" 13 12 

6 1.3 1.2 

7 1.5 1.4 

8 1.7 1.6 

9 2.0 1.8 

10 2.2 2.0 

20 4.3 4.0 

30 6.5 6.0 

40 8.7 8.0 

50 10.8 10.0 

5 

6 

7 

8 

9 

9.84 242 
9.84 255 
9.84 269 
9.84 282 
9.84 295 

9.98 610 
9.98 635 
9.98 661 
9.98 686 
9.98 711 

10.01 390 
10.01 365 
10.01 339 
10.01 314 
10.01 289 

9.85 632 
9.85 620 
9.85 608 
9.85 596 
9.85 583 

55 

54 

53 

52 

51 

10 

11 

12 

13 

14 

9.84 308 
9.84 321 
9.84 334 
9.84 347 
9.84 360 

9.98 737 
9.98 762 
9.98 787 
9.98 812 
9.98 838 

10.01 263 
10.01 238 
10.01 213 
10.01 188 
10.01 162 

9.85 571 
9.85 559 
9.85 547 
9.85 534 
9.85 522 

50 

49 

48 

47 

46 

15 

16 

17 

18 
19 

9.84 373 
9.84 385 
9.84 398 
9.84 411 
9.84 424 

9.98 863 
9.98 888 
9.98 913 
9.98 939 
9.98 964 

10.01 137 
10.01 112 
10.01 087 
10.01 061 
10.01 036 

9.85 510 
9.85 497 
9.85 485 
9.85 473 
9.85 460 

45 

44 

43 

42 

41 

20 

21 

22 

23 

24 

9.84 437 
9.84 450 
9.84 463 
9.84 476 
9.84 489 

9.98 989 

9.99 015 
9.99 040 
9.99 065 
9.99 090 

10.01 011 
10.00 985 
10.00 960 
10.00 935 
10.00 910 

9.85 448 
9.85 436 
9.85 423 
9.85 411 
9.85 399 

40 

39 

38 

37 

36 

25 

26 

27 

28 
29 

9.84 502 
9.84 515 
9.84 528 
9.84 540 
9.84 553 

9.99 116 
9.99 141 
9.99 166 
9.99 191 
9.99 217 

10.00 884 
10.00 859 
10.00 834 
10.00 809 
10.00 783 

9.85 386 
9.85 374 
9.85 361 
9.85 349 
9.85 337 

35 

34 

33 

32 

31 

30 

31 

32 

33 

34 

9.84 566 
9.84 579 
9.84 592 
9.84 605 
9.84 618 

9.99 242 
9.99 267 
9.99 293 
9.99 318 
9.99 343 

10.00 758 
10.00 733 
10.00 707 
10.00 682 
10.00 657 

9.85 324 
9.85 312 
9.85 299 
9.85 287 
9.85 274 

30 

29 

28 

27 

26 

35 

36 

37 

38 

39 

9.84 630 
9.84 643 
9.84 656 
9.84 669 
9.84 682 

9.99 368 
9.99 394 
9.99 419 
9.99 444 
9.99 469 

10.00 632 
10.00 606 
10.00 581 
10.00 556 
10.00 531 

9.85 262 
9.85 250 
9.85 237 
9.85 225 
9.85 212 

25 

24 

23 

22 

21 

40 

41 

42 

43 

44 

9.84 694 
9.84 707 
9.84 720 
9.84 733 
9.84 745 

9.99 495 
9.99 520 
9.99 545 
9.99 570 
9.99 596 

10.00 505 
10.00 480 
10.00 455 
10.00 430 
10.00 404 

9.85 200 

9.85 187 
9.85 175 
9.85 162 
9.85 150 

20 

19 

18 

17 

16 

45 

46 

47 

48 

49 

9.84 758 
9.84 771 
9.84 784 
9.84 796 
9.84 809 

9.99 621 
9.99 646 
9.99 672 
9.99 697 
9.99 722 

10.00 379 
10.00 354 
10.00 328 
10.00 303 
10.00 278 

9.85 137 

9.85 125 
9.85 112 
9.85 100 
9.85 087 

15 

14 

13 

12 

11 

50 

51 

52 

53 

54 

9.84 822 
9.84 835 
9.84 847 
9.84 860 
9.84 873 

9.99 747 
9.99 773 
9.99 798 
9.99 823 
9.99 848 

10.00 253 
10.00 227 
10.00 202 
10.00 177 
10.00 152 

9.85 074 
9.85 062 
9.85 049 
9.85 037 
9.85 024 

10 

9 

8 

7 

6 

55 

56 

57 

58 

59 

9.84 885 
9.84 898 
9.84 911 
9.84 923 
9.84 936 

9.99 874 
9.99 899 
9.99 924 
9.99 949 
9.99 975 

10.00 126 
10.00 101 
10.00 076 
10.00 051 
10.00 025 

9.85 012 

9.84 999 
9.84 986 
9.84 974 
9.84 961 

5 

4 

3 

2 

1 

60 

9.84 949 

0.00 000 

10.00 000 

9.84 949 

0 


L Cos 

d 

L Cot 

c d 

L Tan 

L Sin 

d 

' 

Prop. Pts. 


45 ' 

















































































\ 








TABLE VII 


COMMON LOGARITHMS 
OF NUMBERS 
FROM 

1 TO 10000 

TO 

FIVE DECIMAL PLACES 

1-100 


N 

Log 

N 

Log 

N 

Log 

N 

Log 

N 

Log 

0 

— 

20 

1. 30 103 

40 

1. 60 206 

60 

1. 77 815 

80 

1. 90 309 

1 

0. 00 000 

21 

1. 32 222 

41 

1. 61 278 

61 

1. 78 533 

81 

1. 90 849 

2 

0. 30 103 

22 

1. 34 242 

42 

1. 62 325 

. 62 

1. 79 239 

82 

1. 91 381 

3 

0. 47 712 

23 

1. 36 173 

43 

1. 63 347 

63 

1. 79 934 

83 

1. 91 908 

4 

0. 60 206 

24 

1. 38 021 

44 

1. 64 345 

64 

1. 80 618 

84 

1. 92 428 

5 

0. 69 897 

25 

1. 39 794 

45 

1. 65 321 

65 

1. 81 291 

85 

1. 92 942 

6 

0. 77 815 

26 

1. 41 497 

46 

1. 66 276 

66 

1. 81 954 

86 

1. 93 450 

7 

0. 84 510 

27 

1. 43 136 

47 

1. 67 210 

67 

1. 82 607 

87 

1. 93 952 

8 

0. 90 309 

28 

1. 44 716 

48 

1. 68 124 

68 

1. 83 251 

88 

1. 94 448 

9 

0. 95 424 

29 

1. 46 240 

49 

1. 69 020 

69 

1. 83 885 

89 

1. 94 939 

10 

1. 00 000 

30 

1. 47 712 

50 

1. 69 897 

70 

1. 84 510 

90 

1. 95 424 

n 

1. 04 139 

31 

1. 49 136 

51 

1.70 757 

71 

1. 85 126 

91 

1. 95 904 

12 

1. 07 918 

32 

1. 50 515 

52 

1. 71 600 

72 

1. 85 733 

92 

1. 96 379 

13 

1. 11 394 

33 

1. 51 851 

53 

1. 72 428 

73 

1. 86 332 

93 

1. 96 848 

14 

1. 14 613 

34 

1. 53 148 

54 

1. 73 239 

74 

1. 86 923 

94 

1. 97 313 

15 

1. 17 609 

35 

1. 54 407 

55 

1. 74 036 

75 

1. 87 506 

95 

1. 97 772 

16 

1. 20 412 

36 

1. 55 630 

56 

1. 74 819 

76 

1. 88 081 

96 

1. 98 227 

17 

1. 23 045 

37 

1. 56 820 

57 

1. 75 587 

77 

1. 88 649 

97 

1. 98 677 

18 

1 25 527 

38 

1. 57 978 

58 

1. 76 343 

78 

1. 89 209 

98 

1. 99 123 

19 

1. 27 875 

39 

1. 59 106 

59 

1. 77 085 

79 

1. 89 763 

99 

1. 99 564 

20 

1. 30 103 

40 

1. 60 206 

60 

1. 77 815 

80 

1. 90 309 

100 

2. 00 000 


73 




























74 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


100-150 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop. Pts. 

100 

00 000 

043 

087 

130 

173 

217 

260 

303 

346 

389 





101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 





102 

860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 


44 

43 

42 

103 

01 284 

326 

368 

410 

452 

494 

536 

578 

620 

662 


4.4 

8.8 


104 

703 

745 

787 

828 

870 

912 

953 

995 

*036 

*078 

1 

2 

4.3 

8.6 

4.2 

8.4 

105 

02 119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

3 

4 

13.2 
17 6 

12.9 
17 2 

12.6 

16 8 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

5 

22.0 

21.5 

21.0 

107 

938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

6 

26.4 

25.8 

25.2 

108 

03 342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

7 

30.8 

30.1 

29.4 

109 

743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

8 

9 

35.2 

39.6 

34.4 

38.7 

33.6 

37.8 

110 

04 139 

179 

218 

258 

297 

336 

376 

415 

454 

493 





111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 


41 

40 

39 

112 

922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 


113 

05 308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

1 

2 

4.1 

8.2 

4.0 

8.0 

q q 

114 

690 

729 

767 

805 

843 

«81 

918 

956 

994 

*032 

o.y 

7.8 












3 

12.3 

12.0 

11.7 

115 

06 070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

4 

16.4 

16.0 

15.6 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 

20.5 

20.0 

19.5 

117 

819 

856 

893 

930 

967 

*004 

*041 

*078 

*115 

*151 

6 

24.6 

24.0 

23.4 

118 

07 188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

7 

8 

28.7 
32 8 

28.0 
32 0 

27.3 

31 2 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

9 

36^9 

36i0 

35 ! 1 

120 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 





121 

08 279 

314 

350 

386 

422 

458 

493 

529 

565 

600 


OQ 

37 

36 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 


30 


123 

991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

1 

3.8 

3.7 

3.6 

124 

09 342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

2 

7.6 

7.4 

7.2 












3 

11.4 

11.1 

10.8 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*003 

4 

15.2 
19.0 
22 8 

14.8 
18.5 
22 2 

14.4 

18.0 

21 6 

126 

10 037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

g 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

7 

26!6 

25^9 

25!2 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

8 

30.4 

29.6 

28.8 

129 

11 059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

9 

34.2 

33.3 

32.4 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 





131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 


35 

34 

33 

132 

12 057 

090 

123 

156 

189 

222 

254 

287 

320 

352 





133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

1 

3.5 

3.4 

3.3 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*001 

2 

3 

7.0 

10.5 

6.8 

10.2 

6.6 

9.9 

135 

13 033 

066 

098 

130 

162 

194 

226 

258 

290 

322 

4 

5 

14.0 

17.5 

13.6 

17.0 

13.2 

16.5 

136 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

6 

21.0 

20.4 

19.8 

137 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

7 

24.5 

23.8 

23.1 

138 

988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

8 

28.0 

31.5 

27.2 

30.6 

26.4 

29.7 

139 

14 301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

9 

140 

613 

644 

675 

706 

737 

768 

799 

~ 829 ~ 

860 

891 



31 


141 

922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 


32 

30 

142 

15 229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

1 

2 

3.2 
6 4 

3.1 
6 2 

3.0 

6 0 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

144 

836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

3 

9^6 

9.3 

9.0 












4 

12.8 

12.4 

12.0 

145 

16 137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

5 

16.0 

15.5 

15.0 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

6 

*7 

19.2 
22.4 
25 6 

18.6 
21.7 
24 8 

18.0 

21.0 

24 0 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

/ 

s 

148 

17 026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

9 

28^8 

27.9 

27.0 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 


150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 





N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 



















































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


75 


150-200 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 Prop. Pts. 

150 

17 609 

638 

667 

696 

725 

754 

782 

811 

840 

869 




151 

898 

926 

955 

984 

*013 

*041 

*070 

*099 

*127 

*156 




152 

18 184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

29 

28 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 




154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 1 

2.9 

2.8 











2 

5.8 

5.6 

155 

19 033 

061 

089 

117 

145 

173 

201 

229 

257 

285 I 

8.7 
11 6 

8.4 

11 2 

156 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 I 

14^5 

14 !o 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 6 

17.4 

16.8 

158 

866 

893 

921 

948 

976 

*003 

*030 

*058 

*085 

*112 7 

20.3 

19.6 

159 

20 140 

167 

194 

222 

249 

276 

303 

330 

358 

385 | 

23.2 

26.1 

22.4 

25.2 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 




161 

683 

710 

737 

763 

790 

817 

844 

871 

898 

“ 925 " 




162 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

27 

26 

163 

21 219 

245 

272 

299 

325 

352 

378 

405 

431 

458 , 

2 7 

2 6 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 2 

5.4 

5.2 











3 

8.1 

7.8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 4 

10.8 

10.4 

166 

22 011 

037 

063 

089 

115 

141 

167 

194 

220 

246 5 

13.5 

13.0 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 ^ 

16.2 
1 fi Q 

15.6 

1 S 9 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 l 

lo.y 

21.6 

10.4 

20.8 

169 

789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 9 

24.3 

23.4 

170 

23 045 

070 

096 

121 

147 

172 

198 

223 

249 

274 




171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 


2 H 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 




173 

805 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 

1 

2.5 

174 

24 055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

2 

5.0 












3 

7.5 

175 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4 

5 

10.0 

12 5 

176 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 ; 

6 

15 !o 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 

7 

17.5 

178 

25 042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

8 

20.0 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 ' 

9 

22.5 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 




181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

23 

182 

26 007 

031 

055 

079 

102 

126 

150 

174 

198 

221 




183 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 l 

Z.'i 
A 8 

2.3 

4 6 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 § 

rt.O 

7.2 

6i9 











4 

9.6 

9.2 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 5 

12.0 

11.5 

186 

951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 6 

14.4 

13.8 

1 A 1 

187 

27 184 

207 

231 

254 

277 

300 

323 

346 

370 

393 l 

16.8 
19 2 

ID. 1 

18 4 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 q 

21*6 

20^7 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 




190 

875 

898 

921 

944 

967 

989 

*012 

*035 

*058 

*081 


O 

21 

191 

28 103 

126 

149 

171 

194 

217 

240 

262 

285 

307 


192 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 i 

2.2 

2.1 

193 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 2 

4.4 

4.2 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 3 

4 

6.6 

8.8 

6.3 

8.4 

195 

29 003 

026 

048 

070 

092 

115 

137 

159 

181 

203 6 

11.0 

13.2 

10.5 

12.6 

196 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 7 

15.4 

14.7 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 8 

17.6 

16.8 

198 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 9 

19.8 

18.9 

199 

885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 




200 

30 103 

125 

146 

168 

190 

211 

233 

255 

276 

298 




N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 Prop. Pts. 












































































76 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


200-250 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

200 

30 103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

22 21 

1 2.2 2.1 

2 4.4 4.2 

3 6.6 6.3 

4 8.8 8.4 

5 11.0 10.5 

6 13.2 12.6 

7 15.4 14.7 

8 17.6 16.8 

9 19.8 18.9 

20 

1 2.0 

2 4.0 

3 6.0 

4 8.0 

5 10.0 

6 12.0 

7 14.0 

8 16.0 

9 18.0 

19 

1 1.9 

2 3.8 

3 5.7 

4 7.6 

5 9.5 

6 11.4 

7 13.3 

8 15.2 

9 17.1 

18 

1 1.8 

2 3.6 

3 5.4 

4 7.2 

5 9.0 

6 10.8 

7 12.6 

8 14.4 

9 16.2 

17 

1 1.7 

2 3.4 

3 5.1 

4 6.8 

5 8.5 

6 10.2 

7 11.9 

8 13.6 

9 15.3 

201 

202 

203 

204 

205 

206 

207 

208 
209 

320 

535 

750 

963 

31 175 
387 
597 
806 

32 015 

341 

557 

771 

984 

197 

408 

618 

827 

035 

363 

578 

792 

*006 

218 

429 

639 

848 

056 

384 

600 

814 

*027 

239 

450 

660 

869 

077 

406 

621 

835 

*048 

260 

471 

681 

890 

098 

428 

643 

856 

*069 

281 

492 

702 

911 

118 

449 

664 

878 

*091 

302 

513 

723 

931 

139 

471 

685 

899 

*112 

323 
534 
744 
952 1 
160 

492 

707 

920 

*133 

345 

555 

765 

973 

181 

514 

728 

942 

*154 

366 

576 

785 

994 

201 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

211 

212 

213 

214 

215 

216 

217 

218 
219 

428 

634 

838 

33 041 

244 

445 

646 

846 

34 044 

449 

654 

858 

062 

264 

465 

666 

866 

064 

469 

675 

879 

082 

284 

486 

686 

885 

084 

490 

695 

899 

102 

304 

506 

706 

905 

104 

510 

715 

919 

122 

325 

526 

726 

925 

124 

531 

736 

940 

143 

345 

546 

746 

945 

143 

552 

756 

960 

163 

365 

566 

766 

965 

163 

572 
777 
980 i 
183 

385 

586 

786 

985 

183 

593 

797 

*001 

203 

405 

606 

806 

*005 

203 

613 

818 

*021 

224 

425 

626 

826 

*025 

223 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

221 

222 

223 

224 

225 

226 

227 

228 
229 

439 
635 
830 
35 025 

218 

411 

603 

793 

984 

459 

655 

850 

044 

238 

430 

622 

813 

*003 

479 

674 

869 

064 

257 

449 

641 

832 

*021 

498 

694 

889 

083 

276 

468 

660 

851 

*040 

518 

713 

908 

102 

295 

488 

679 

870 

*059 

537 

733 

928 

122 

315 

507 

698 

889 

*078 

557 

753 

947 

141 

334 

526 

717 

908 

*097 

577 

772 

967 

160 

353 

545 

736 

927 

*116 

596 

792 

986 

180 

372 

564 

755 

946 

*135 

616 

811 

*005 

199 

392 

583 

774 

965 

*154 

230 

36 173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

231 

232 

233 

234 

235 

236 

237 

238 

239 

361 

549 

736 

922 

37 107 
291 
475 
658 
840 

380 

568 

754 

940 

125 

310 

493 

676 

858 

399 

586 

773 

959 

144 

328 

511 

694 

876 

418 

605 

791 

977 

162 

346 

530 

712 

894 

436 

624 

810 

996 

181 

365 

548 

731 

912 

455 

642 

829 

*014 

199 

383 

566 

749 

931 

474 

661 

847 

*033 

218 

401 

585 

767 

949 

493 

680 

866 

*051 

236 

420 

603 

785 

967 

511 

698 

884 

*070 

254 

438 

621 

803 

985 

530 

717 

903 

*088 

273 

457 

639 

822 

*003 

240 

38 021 

039 

057 

075 

093 

112 

130 

148 

! 166 

184 

241 

242 

243 

244 

245 

246 

247 

248 

249 

202 

382 

561 

739 

917 
39 094 
270 
445 
620 

220 

399 

578 

757 

934 

111 

287 

463 

637 

238 

417 

596 

775 

952 

129 

305 

480 

655 

256 

435 

614 

792 

970 

146 

322 

498 

672 

274 

453 

632 

810 

987 

164 

340 

515 

690 

292 

471 

650 

828 

*005 

182 

358 

533 

707 

310 

489 

668 

846 

*023 

199 

375 

550 

724 

328 

507 

686 

863 

*041 

217 

393 

568 

742 

346 

525 

703 

881 

*058 

235 

410 

585 

759 

364 

543 

721 

899 

*076 

252 

428 

602 

777 

250 

794 

811 

829 

^846 

863 

881 

898 

915 

933 

950 

N 

L 0 

1 

2 3 

4 

5 

6 

7 

1 8 

9 

Prop. Pts. 












































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


77 


250-300 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

250 

39 794 

811 

829 

846 

863 

881 

898 

915 

933 

950 



251 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 



252 

40 140 

157 

175 

192 

209 

226 

243 

261 

278 

295 


18 

253 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 


254 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

1 

1.8 











2 

3.6 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

3 

5.4 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

4 

5 

7.2 

9.0 

10.8 

257 

993 

*010 

*027 

*044 

*061 

*078 

*095 

*111 

*128 

*145 

6 

258 

41 162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

7 

12.6 

259 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

8 

14.4 










9 

16.2 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 


261 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 



262 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 


17 

263 

996 

*012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

*144 

1 

2 

1.7 

3.4 

264 

42 160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

265 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

3 

4 

5.1 

6.8 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

5 

8.5 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

6 

10.2 

268 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

7 

Q 

11.9 

13.6 

15.3 

269 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

o 

9 

270 

43 136 

152 

169 

185 

201 

217 

233 

249 

265 

281 



271 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 


16 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 


273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

1 

1.6 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

2 

3.2 












3 

4.8 

275 

933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

4 

6.4 

8.0 

9 6 

276 

44 091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

o 

6 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

7 

11.2 

278 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

8 

12.8 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

9 

14.4 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 



281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 


15 

282 

45 025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

1 

o 

1.5 

3.0 

4.5 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 












4 

6.0 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

5 

7.5 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

6 

9.0 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

7 

8 

9 

10.5 

12.0 

13.5 

288 

939 

954 

969 

984 

*000 

*015 

*030 

*045 

*060 

*075 

289 

46 090 

105 

120 

135 

150 

165 

180 

195 

210 

225 


290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 


14 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 


292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

1 

1.4 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

2 

2.8 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

3 

4.2 









4 

5.6 

295 

982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

*114 

5 

g 

7.0 

8 4 

296 

47 129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

7 

9i8 

297 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

8 

11.2 

298 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

9 

12.6 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 



300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 


log c = .43429 




























































78 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


300-350 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

300 

47 712 

727 

741 

756 

770 

784 

799 

813 

828 

842 



*301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 



302 

48 001 

015 

029 

044 

058 

073 

087 

101 

116 

130 



303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 



304 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 


15 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

1 

1.5 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

2 

Q 

3.0 

4.5 

6.0 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

o 

4 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

5 

7.5 

309 

996 

*010 

*024 

*038 

*052 

*066 

*080 

*094 

*108 

*122 

6 

9.0 










7 

10.5 

310 

49 136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

8 

9 

12.0 

13.5 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 



312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 



313 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 



314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 



315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 


14 

316 

969 

982 

996 

*010 

*024 

*037 

*051 

*065 

*079 

*092 


317 

50 106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

1 

1.4 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

2 

2.8 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

3 

4 

4.2 

5.6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

5 

6 

7.0 

8.4 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9.8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

Q 

11.2 

12.6 

323 

920 

934 

947 

961 

974 

987 

*001 

*014 

*028 

*041 

y 

324 

51 055 

068 

081 

095 

108 

121 

135 

148 

162 

175 



325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 



326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 



327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 


13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 


329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

1 

2 

1.3 

2 6 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

3 

3.9 

5.2 

6.5 

331 

983 

996 

*009 

*022 

*035 

*048 

*061 

*075 

*088 

*101 

5 

332 

52 114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

6 

7.8 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

7 

9.1 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

8 

9 

10.4 

11.7 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 



336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 



337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 



338 

892 

905 

917 

930 

943 

956 

969 

982 

994 

*007 



339 

53 020 

033 

046 

058 

071 

084 

097 

110 

122 

135 


12 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

1 

1.2 

341 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

2 

2.4 

342 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

3 

3.6 

343 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

4 

K 

4.8 

6.0 

7.2 

8.4 

9.6 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

o 

6 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

7 

8 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*008 

*020 

9 

10.8 

347 

54 033 

045 

058 

070 

083 

095 

108 

120 

133 

145 



348 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 



349 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 



350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 


log 7 r = .49715 













































































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


79 


350-400 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

350 

54 407 

419 

432 

444 

456 

469 

481 

494 

506 

518 



351 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 



352 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 



353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 



354 

900 

913 

925 

937 

949 

962 

974 

986 

998 

*011 


13 

355 

55 023 

035 

047 

060 

072 

084 

096 

108 

121 

133 

1 

2 

1.3 

2 6 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

3 

3.9 

357 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

4 

5.2 

358 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

5 

6.5 

359 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

6 

7 

7.8 

9.1 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

8 

9 

10.4 

11.7 

361 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 



362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 



363 

991 

*003 

*015 

*027 

*038 

*050 

*062 

*074 

*086 

*098 



364 

56 110 

122 

134 

146 

158 

170 

182 

194 

205 

217 



365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 


12 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 



367 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

1 

1.2 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

2 

3 

2.4 

3 6 

369 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

4 

c 

4^8 

A A 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

O 

6 

O.U 

7.2 

371 

937 

949 

961 

972 

984 

996 

*008 

*019 

*031 

*043 

7 

g 

8.4 

9 6 

372 

57 054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

9 

10.8 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 



374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 



375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 



376 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 



377 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 


11 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 



379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

1 

1.1 












2 

2.2 

380 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

3 

4 

3.3 

4 4 

381 

58 092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

5 

5.5 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

6 

6.6 

383 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

7 

7.7 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

8 

9 

8.8 

9.9 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 



386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 



387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 



388 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 



389 

995 

*006 

*017 

*028 

*040 

*051 

*062 

*073 

*084 

*095 


10 

390 

59 106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

1 

1.0 

391 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

2 

2.0 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

3 

A 

3.0 

A rt 

393 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

5 

5 0 

394 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 

6 

6i0 












7 

7.0 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

8 

8.0 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

9 

9.0 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 



398 

988 

999 

*010 

*021 

*032 

*M3 

*054 

*065 

*076 

*086 



399 

60 097 

108 

119 

130 

141 

152 

163 

173 

184 

195 



400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 




















































80 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


400-450 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

S 

9 

Prop. Pts. 

400 

60 206 

217 

228 

239 

249 

260 

271 

282 

293 

304 



401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 



402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 



403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 



404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 



405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 



406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 


11 

407 

959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 



408 

61 066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

1 

1.1 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 

3 

2.2 

3.3 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

4 

5 

4.4 

5.5 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

6 

6.6 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

7 

7.7 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

8 

Q 

8.8 

Q G 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

y 

y.y 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 



416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 



417 

62 014 

024 

034 

045 

055 

066 

076 

086 

097 

107 



418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 



419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 



420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 



421 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 


10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 



423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

i 

1.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

2 

2.0 












3 

3.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

4 

K 

4.0 

5 0 

426 

941 

951 

961 

972 

982 

992 

*002 

*012 

*022 

*033 

u 

6 

6!o 

427 

63 043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

7 

7.0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

8 

8.0 

429 

246 

256 

266 

276 

28Q 

296 

306 

317 

327 

337 

9 

9.0 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 



431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 



432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 



433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 



434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 



435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 



436 

949 

959 

969 

979 

988 

998 

*008 

*018 

*028 

*038 



437 

64 048 

058 

068 

078 

088 

098 

108 

118 

128 

137 


9 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

1 

0 9 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

2 

1.8 












Q 

2.7 












O 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

4 

3.6 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

5 

5 

4.5 

5 4 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

7 

6^3 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

8 

7.2 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

9 

8.1 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 



446 

933 

943 

953 

963 

972 

982 

992 

*002 

*011 

*021 



447 

65 031 

040 

050 

060 

070 

079 

089 

099 

108 

118 



448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 



449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 



450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 














































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


81 


450-500 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

450 

65 321 

331 

341 

350 

360 

369 

379 

389 

398 

408 



451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 



452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 



453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 



454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 



455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 



456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 


10 

457 

992 

*001 

*011 

*020 

*030 

*039 

*049 

*058 

*068 

*077 



458 

66 087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

1 

1.0 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

2 

2.0 

3.0 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

4 

5 

4.0 

5.0 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

6 

6.0 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

7 

7.0 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

8 

Q 

8.0 
q n 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

y 

y.u 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 



466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 



467 

932 

941 

950 

960 

969 

978 

987 

997 

*006 

*015 



468 

67 025 

034 

043 

052 

062 

071 

080 

089 

099 

108 



469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 



470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 



471 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 


g 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 



473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

1 

0.9 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

2 

1.8 












3 

2.7 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

4 

5 

3.6 

4 5 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

6 

5^4 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

7 

6.3 

478 

943 

952 

961 

970 

979 

988 

997 

*006 

*015 

*024 

8 

7.2 

479 

68 034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

9 

8.1 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 



481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 



482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 



483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 



484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 



485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 



486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 



487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 


8 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

1 

0.8 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*011 

2 

1.6 












3 

2 4 

490 

69 020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

4 

r 

3^2 

A n 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

D 

6 

4.U 

4.8 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

7 

5.6 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

8 

6.4 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

9 

7.2 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 



496 

548 

557 

556 

574 

583 

592 

601 

609 

618 

627 



497 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 



498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 



499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 



500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 



























































82 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


500-550 


JN 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

500 

69 897 

906 

914 

923 

932 

940 

949 

958 

966 

975 



501 

984 

992 

*001 

*010 

*018 

*027 

*036 

*044 

*053 

*062 



502 

70 070 

079 

088 

096 

105 

114 

122 

131 

140 

148 



503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 



504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 



505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 



506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 


9 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 



508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

1 

0.9 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

2 

1.8 









3 

2.7 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

4 

5 

3.6 

4.5 

5.4 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

6 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

*003 

7 

6.3 

513 

71 012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

8 

7.2 

514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

9 

8.1 

515 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 



516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 



517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 



518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 



519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 



520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 



521 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 



522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 


8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

1 

0.8 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

2 

1.6 












3 

2.4 

525 

72 016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

4 

3.2 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4.0 

4.8 

5.6 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

U 

7 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

8 

6.4 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

9 

7.2 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 



531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 



532 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 



533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 



534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 



535 

835 

843' 

852 

860 

868 

876 

884 

892 

900 

908 



536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 



537 

997 

*006 

*014 

*022 

*030 

*038 

*046 

*054 

*062 

*070 


7 

538 

73 078 

086 

094 

102 

111 

119 

127 

135 

143 

151 


0.7 

1.4 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

1 

2 

3 

4 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

2.1 

2.8 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

5 

3.5 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

6 

7 

4.2 

4.9 

5 6 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

# 

8 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

9 

6.3 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 



546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 



547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 



548 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 



549 

957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 



550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 





































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


83 


550-600 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 



552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 



553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 



554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 



555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 



556 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 



557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 



558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 



559 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 



560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 



561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 


8 

562 

974 

981 

989 

997 

*005 

*012 

*020 

*028 

*035 

*043 


0.8 

1.6 

2.4 

563 

75 051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

1 

o 

564 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

& 

3 












4 

3.2 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

5 

4.0 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

6 

4.8 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

7 

8 
g 

5.6 

6.4 

7.2 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 


570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 



571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 



572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 



573 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 



574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 



575 

967 

074 

982 

989 

997 

*005 

*012 

*020 

*027 

*035 



576 

76 042 

050 

057 

065 

072 

080 

087 

095 

103 

110 



577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 



578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 



579 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 



580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 



581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 



582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 


7 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

1 

2 

0 7 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

1.4 












3 

2.1 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

4 

2.8 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

5 

6 

7 

3.5 

4.2 

4 9 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

588 

938 

945 

953 

960 

867 

975 

982 

989 

997 

*004 

8 

5.6 

589 

77 012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

9 

6.3 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 



591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 



592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 



593 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 



594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 



595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 



596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 



597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 



598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 



599 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 



600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

! 8 

9 

Prop. Pts. 


























































































84 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


600-650 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

600 

77 815 

822 

830 

837 

844 

851 

859 

866 

873 

880 



601 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 



602 

960 

967 

974 

981 

988 

996 

*003 

*010 

*017 

*025 



603 

78 032 

039 

046 

053 

061 

068 

075 

082 

089 

097 



604 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 



605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 



606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 


8 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 


0.8 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

1 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

2 

3 

1.6 

2.4 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

4 

5 

3.2 

4.0 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

6 

4.8 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

7 

8 
g 

5.6 

613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

7^2 

614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 


615 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 



616 

958 

965 

972 

979 

986 

993 

*000 

*007 

*014 

*021 



617 

79 029 

036 

043 

050 

057 

064 

071 

078 

085 

092 



618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 



619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 



620 

239 

246 

253 

260 

267. 

274 

281 

288 

295 

302 



621 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 


7 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 



623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

1 

0.7 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

2 

3 

1.4 

2.1 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

. 4 

5 

2.8 

3 5 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

6 

4.2 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

7 

4.9 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

8 

5.6 

629 

865 

872 

879 

886 

893 

.900 

906 

913 

920 

927 

9 

6.3 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 



631 

80 003 

010 

017 

024 

030 

037 

044 

051 

058 

065 



632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 



633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 



634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 



635 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 



636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 



637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 


6 

638 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

1 

0.6 

639 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

2 

3 

1.2 

1.8 

2.4 

3.0 

3.6 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

4 

e 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

O 

6 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

7 

4.2 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

8 

4.8 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

9 

5.4 

645 

956 

963 

969 

976 

983 

990 

996 

*003 

*010 

*017 



646 

81 023 

030 

037 

043 

050 

057 

064 

070 

077 

084 



647 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 



648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 



649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 



650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 





































































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


85 


650-700 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

650 

81 291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

7 

1 0.7 

2 1.4 

3 2.1 

4 2.8 

5 3.5 

6 4.2 

7 4.9 

8 5.6 

9 6.3 

6 

1 0.6 

2 1.2 

3 1.8 

4 2.4 

5 3.0 

6 3.6 

7 4.2 

8 4.8 

9 5.4 

651 

652 

653 

654 

655 

656 

657 

658 

659 

358 

425 

491 

558 

624 

690 

757 

823 

889 

365 

431 

498 

564 

631 

697 

763 

829 

895 

371 

438 

505 

571 

637 

704 

770 

836 

902 

378 

445 

511 

578 

644 

710 

776 

842 

908 

385 

451 

518 

584 

651 

717 

783 

849 

915 

391 

458 

525 

591 

657 

723 

790 

856 

921 

398 

465 

531 

598 

664 

730 

796 

862 

928 

405 

471 

538 

604 

671 

737 

803 

869 

935 

411 

478 

544 

611 

677 

743 

809 

875 

941 

418 

485 

551 

617 

684 

750 

816 

882 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 

661 

662 

663 

664 

665 

666 

667 

668 
669 

82 020 
086 
151 
217 

282 

347 

413 

478 

543 

027 

092 

158 

223 

289 

354 

419 

484 

549 

033 

099 

164 

230 

295 

360 

426 

491 

556 

040 

105 

171 

236 

302 

367 

432 

497 

562 

046 

112 

178 

243 

308 

373 

439 

504 

569 

053 

119 

184 

249 

315 

380 

445 

510 

575 

060 

125 

191 

256 

321 

387 

452 

517 

582 

066 

132 

197 

263 

328 

393 

458 

523 

588 

073 

138 

204 

269 

334 

400 

465 

530 

595 

079 

145 

210 

276 

341 

406 

471 

536 

601 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672 

673 

674 

675 

676 

677 

678 

679 

672 

737 

802 

866 

930 
995 
83 059 
123 
187 

679 

743 

808 

872 

937 

*001 

065 

129 

193 

685 

750 

814 

879 

943 

*008 

072 

136 

200 

692 

756 

821 

885 

950 

*014 

078 

142 

206 

698 

763 

827 

892 

956 

*020 

085 

149 

213 

705 

769 

834 

898 

963 

*027 

091 

155 

219 

711 

776 

840 

905 

969 

*033 

097 

161 

225 

718 

782 

847 

911 

975 

*040 

104 

168 

232 

724 

789 

853 

918 

982 

*046 

110 

174 

238 

730 

795 

860 

924 

988 

*052 

117 

181 

245 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 

682 

683 

684 

685 

686 

687 

688 
689 

315 

378 

442 

506 

569 

632 

696 

759 

822 

321 

385 

448 

512 

575 

639 

702 

765 

828 

327 

391 

455 

518 

582 

645 

708 

771 

835 

334 

398 

461 

525 

588 

651 

715 

778 

841 

340 

404 

467 

531 

594 

658 

721 

784 

847 

347 

410 

474 

537 

601 

664 

727 

790 

853 

353 

417 

480 

544 

607 

670 

734 

797 

860 

359 

423 

487 

550 

613 

677 

740 

803 

866 

366 

429 

493 

556 

620 

683 

746 

809 

872 

372 

436 

499 

563 

626 

689 

753 

816 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

692 

693 

694 

695 

696 

697 

698 

699 

948 
84 011 
073 
136 

198 

261 

323 

386 

448 

954 

017 

080 

142 

205 

267 

330 

392 

454 

960 

023 

086 

148 

211 

273 

336 

398 

460 

967 

029 

092 

155 

217 

280 

342 

404 

466 

973 

036 

098 

161 

223 

286 

348 

410 

473 

979 

042 

105 

167 

230 

292 

354 

417 

479 

985 

048 

111 

173 

236 

298 

361 

423 

485 

992 

055 

117 

180 

242 

305 

367 

429 

491 

998 

061 

123 

186 

248 

311 

373 

435 

497 

*004 

067 

130 

192 

255 

317 

379 

442 

504 


700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 





































































































































86 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


700-750 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

700 

84 510 

516 

522 

528 

535 

541 

547 

553 

559 

566 



701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 



702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 



703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 



704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 



705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 



706 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 


7 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 



708 

85 003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

1 

0.7 

709 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

2 

3 

1.4 

2.1 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

4 

5 

2.8 

3.5 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

6 

4.2 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

7 

4.9 

713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

8 

Q 

5.6 

6.3 

714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

y 

715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 



716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 



717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 



718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 



719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 



720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 



721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 


c 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 


D 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

i 

0.6 

724 

974 

980 

986 

992 

998 

*004 

*010 

*016 

*022 

*028 

2 

1.2 












3 

1.8 

725 

86 034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

4 

K 

2.4 

3.0 

3 6 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

o 

6 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

7 

4.2 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

8 

4.8 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9 

5.4 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 



731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 



732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 



733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 



734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 



735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 



736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 



737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 


5 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

1 

2 

Q 

0 5 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

lio 

1.5 

2.0 

2.5 

3.0 

3.5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

O 

4 

741 

982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

5 

5 

742 

87 040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

7 

743 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

8 

4.0 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

9 

4.5 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 



746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 



747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 



748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 



749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 



750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 
















































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


87 


750-800 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

750 

87 506 

512 

518 

523 

529 

535 

541 

547 

552 

558 



751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 



752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 



753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 



754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 



755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 



756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 



757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 



758 

967 

973 

978 

984 

990 

996 

*001 

*007 

*013 

*018 



759 

88 024 

030 

036 

041 

047 

053 

058 

064 

070 

076 



760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 



761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 


6 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 


0.6 

1 2 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

1 

2 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

3 

L8 












4 

2.4 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

5 

3.0 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

6 

3.6 

A O 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

7 

g 

4. L 

4 8 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

9 

5 I 4 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 



770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 



771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 



772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 



773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 



774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 



775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 



776 

986 

992 

997 

*003 

*009 

*014 

*020 

*025 

*031 

*037 



777 

89 042 

048 

053 

059 

064 

070 

076 

081 

087 

092 



778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 



779 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 



780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 



781 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 



782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 


5 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

1 

0.5 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

2 

1.0 












3 

1.5 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

4 

2.0 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

5 

0 

2.5 

3 0 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

7 

3^5 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

8 

4.0 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

9 

4.5 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 



791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 



792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 



793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 



794 

982 

988 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 



795 

90 037 

042 

048 

053 

059 

064 

069 

075 

080 

086 



796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 



797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 



798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 



799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 



800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 














































88 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


800-850 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

800 

90 309 

314 

320 

325 

331 

336 

342 

347 

352 

358 



801 

363 

369 

374 

380 

385 

390 

396 

401 

407 

412 



802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 



803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 



804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 



805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 



806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 



807 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 



808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 



809 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 



810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 



811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 


6 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*004 

1 

0 6 

813 

91 009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

l!2 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105 

110 

3 

1.8 












4 

2.4 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

5 

3.0 

816 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

6 

7 

3.6 

4 2 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

l 

8 

4!8 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

9 

5.4 

819 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 



820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 



821 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 



822 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 



823 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 



824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 



825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 



826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 



827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 



828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 



829 

855 

861 

866 

871 

876. 

882 

887 

892 

897 

903 



830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 



831 

960 

965 

971 

976 

981 

986 

991 

997 

*002 

*007 



832 

92 012 

018 

023 

028 

033 

038 

044 

049 

054 

059 


5 

833 

065 

070 

075 

080 

085 

091 

096 

101 

106 

111 

1 

0.5 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

2 

1.0 












3 

1.5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

4 

2.0 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

« 

2.5 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

U 

7 

O.U 

3.5 

838 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

8 

4.0 

839 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

9 

4.5 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 



841 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 



842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 



843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 



844 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 



845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 



846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 



847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 



848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 



849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 



850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 



































































































VII FIVE-PLACE LOGARITHMS OF NUMBERS 89 

850-900 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

850 

92 942 

947 

952 

957 

962 

967 

973 

978 

983 

988 



851 

993 

998 

*003 

*008 

*013 

*018 

*024 

*029 

*034 

*039 



852 

93 044 

049 

054 

059 

064 

069 

075 

080 

085 

090 



853 

095 

100 

105 

110 

115 

120 

125 

131 

181 

141 



854 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 



855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 



856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 


6 

857 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 



858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

1 

0.6 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

2 

3 

1.2 

1.8 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

4 

5 

2.4 

3.0 

861 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

6 

3.6 

862 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

7 

4.2 

4.8 

5.4 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

8 

g 

864 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 


865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 



866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 



867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 



868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 



869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 



870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 



871 

94 002 

007 

012 

017 

022 

027 

032 

037 

042 

047 


5 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 



873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

1 

0.5 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

2 

3 

1.0 

1.5 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

4 

5 

2.0 

2.5 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

6 

3.0 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

7 

3.5 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

8 

o 

4.0 

4.5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

y 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 



881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 



882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 



883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 



884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 



885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 



886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 


4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 



888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

i 

0.4 

889 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

2 

3 

0.8 

1.2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

4 

5 

1.6 

2.0 

891 

988 

993 

998 

*002 

*007 

*012 

*017 

*022 

*027 

*032 

6 

2.4 

892 

95 036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

7 

2.8 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

8 

3.2 

3.6 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

9 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 



896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 



897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 



898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 



899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 



900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 































































































90 


FIVE-PLACE LOGARITHMS OF NUMBERS 


VII 


900-950 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

900 

95 424 

429 

434 

439 

444 

448 

453 

458 

463 

468 



901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

"516 



902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 



903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 



904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 



905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 



906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 



907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 



908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 



909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 



910 

904 

909 

914 

018 

923 

928 

933 

938 

942 

947 



911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 


5 

912 

999 

*004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 



913 

96 047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

1 

2 

0.5 

1 0 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

3 

1.5 












4 

2.0 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

5 

2.5 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

6 

3.0 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

7 

Q 

3.5 

4.0 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

O 

9 

4^5 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 



920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 



921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 



922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 



923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 



924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 



925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 



926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 



927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 



928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 



929 

802 

806 

811 

816 

820 

825 

830 

834 

S39 

844 



930 

848 

853 

858 

862 

876 

872 

876 

881 

886 

890 



931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 



932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 


4 

933 

988 

993 

997 

*002 

*007 

*011 

*016 

*021 

*025 

*030 


n a 

934 

97 035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

2 

0.8 












3 

1.2 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

4 

1.6 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

2.0 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

6 

2.4 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

8 

3 2 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

9 

3.6 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 



941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 



942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 



943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 



944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 



945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 



946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 



947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 



948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 



949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 



950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 



































VII 


FIVE-PLACE LOGARITHMS OF NUMBERS 


91 


950-1000 


N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 

950 

97 772 

777 

782 

786 

791 

795 

800 

804 

809 

813 



951 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 



952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 



953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 



954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 



955 

98 000 

005 

009 

014 

019 

023 

028 

032 

037 

041 



956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 



957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 



958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 



959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 



960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 



961 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 


5 

962 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 



963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

1 

9 

0.5 

1 n 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

a 

3 

1 .u 

1.5 












4 

2.0 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

5 

2.5 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

6 

3.0 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

7 

g 

3.5 

4.0 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

9 

4^5 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 



970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 



971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 



972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 



973 

811 

816 

•820 

825 

829 

834 

838 

843 

847 

851 



974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 



975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 



976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 



977 

989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 



978 

99 034 

038 

043 

047 

052 

056 

061 

065 

069 

074 



979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 



980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 



981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 



982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 


4 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

1 

0 4 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

2 

0.8 












3 

1.2 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

4 

1.6 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

5 

2.0 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

7 

2 8 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

8 

3.2 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

9 

3.6 

990 

564 

568 

572 

577 

581 

585 

590 

594" 

599 

603 



991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 



992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 



993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 



994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 



995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 



996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 



997 

870 . 

874 

878 

883 

887 

891 

896 

900 

904 

909 



998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

« 


999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 



1000 

00 000 

004 

009 

013 

017 

022 

026 

030 

035 

039 



N 

L 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Pts. 


















































































TABLE VIII 


NATURAL LOGARITHMS OF NUMBERS 
BASE e = 2.71828... 

Note. — Log e 10 N = Log e N + Log« 10 
N 

Log e — = Log e N - Log e 10 
Log e 10 = 2.30259 

For example : Log 27 = Log e 2.7 + Log e 10 

= 0.99325 + 2.30259 = 3.29584 
Log e .27 = Log e 2.7 — Log e 10 

= 0.99325 - 2.30259 = 8.69066 


10 



0 

1 


3 

4 

5 

6 

7 

8 

9 

1.0 

0.0 0000 

0995 

1980 

2956 

3922 

4S79 

5827 

6766 

7696 

8618 

1.1 

9531 

*0436 

*1333 

*2222 

*3103 

*3976 

*4842 

*5700 

*6551 

*7395 

1.2 

0.1 8232 

9062 

9885 

*0701 

*1511 

*2314 

*3111 

*3902 

*4686 

*5464 

1.3 

0.2 6236 

7003 

7763 

8518 

9267 

*0010 

*0748 

*1481 

*2208 

*2930 

1.4 

0.3 3647 

4359 

5066 

5767 

6464 

7156 

7844 

8526 

9204 

9878 

1.5 

0.4 0547 

1211 

1871 

2527 

3178 

3825 

4469 

5108 

5742 

6373 

1.6 

7000 

7623 

8243 

8858 

9470 

*0078 

*0682 

*1282 

*1879 

*2473 

1.7 

0.5 3063 

3649 

4232 

4812 

5389 

5962 

6531 

7098 

7661 

8222 

1.8 

8779 

9333 

9884 

*0432 

*0977 

*1519 

*2078 

*2594 

*3127 

*3658 

1.9 

0.6 4185 

4710 

5233 

5752 

6269 

6783 

7294 

7803 

8310 

8813 

2.0 

9315 

9813 

*0310 

*0804 

*1295 

*1784 

*2271 

*2755 

*3237 

*3716 

2.1 

0.7 4194 

4669 

5142 

5612 

6081 

6547 

7011 

7473 

7932 

8390 

2.2 

8846 

9299 

9751 

*0200 

*0648 

*1093 

*1536 

*1978 

*2418 

*2855 

2.3 

0.8 3291 

3725 

4157 

4587 

5015 

5442 

5866 

6289 

6710 

7129 

2.4 

7547 

7963 

8377 

8789 

9200 

9609 

*0016 

*0422 

*0826 

*1228 

2.5 

0.9 1629 

2028 

2426 

2822 

3216 

3609 

4001 

4391 

4779 

5166 

2.6 

5551 

5935 

6317 

6698 

7078 

7456 

7833 

8208 

8582 

8954 

2.7 

9325 

9695 

*0063 

*0430 

*0796 

*1160 

*1523 

*1885 

*2245 

*2604 

2.8 

1.0 2962 

3318 

3674 

4028 

4380 

4732 

5082 

5431 

5779 

6126 

2.9 

6471 

6815 

7158 

7&00 

7841 

8181 

8519 

8856 

9192 

9527 

3.0 

9861 

*0194 

*0526 

*0856 

*1186 

*1514 

*1841 

*2168 

*2493 

*2817 

3.1 

1.1 3140 

3462 

3783 

4103 

4422 

4740 

5057 

5373 

5688 

6002 

3.2 

6315 

6627 

6938 

7248 

7557 

7865 

8173 

8479 

8784 

9089 

3.3 

9392 

9695 

9996 

*0297 

*0597 

*0896 

*1194 

*1491 

*1788 

*2083 

3.4 

1.2 2378 

2671 

2964 

3256 

3547 

3837 

4427 

4415 

4703 

4990 

3.5 

5276 

5562 

5846 

6130 

6413 

6695 

6976 

7257 

7536 

7815 

3.6 

8093 

8371 

8647 

8923 

9198 

9473 

9746 

*0019 

*0291 

*0563 

3.7 

1.3 0833 

1103 

1372 

1641 

1909 

2176 

2442 

2708 

2972 

3237 

3.8 

3500 

3763 

4025 

4286 

4547 

4807 

5067 

5325 

5584 

5841 

3.9 

6098 

6354 

6609 

6864 

7118 

7372 

7624 

7877 

8128 

8379 

4.0 

8629 

8879 

9128 

9377 

9624 

9872 

*0118 

*0364 

*0610 

*0854 

4.1 

1.4 1099 

1342 

1585 

1828 

2070 

2311 

2552 

2792 

3031 

3270 

4.2 

3508 

3746 

3984 

4220 

4456 

4692 

4927 

5161 

5395 

5629 

4.3 

5862 

6094 

6326 

6557 

6787 

7018 

7247 

7476 

7705 

7933 

4.4 

8160 

8387 

8614 

8840 

9065 

9290 

9515 

9739 

9962 

*0185 

4.5 

1.5 0408 

0630 

0851 

1072 

1293 

1513 

1732 

1951 

2170 

238S 

4.6 

2606 

2823 

3039 

3256 

3471 

3687 

3902 

4116 

4330 

4543 

4.7 

4756 . 

4969 

5181 

5393 

5604 

5814 

6025 

6235 

* 6444 

6653 

4.8 

6862 

7070 

7277 

7485 

7691 

7898 

8104 

8309 

8515 

• 8719 

4.9 

8924 

9127 

9331 

9534 

9737 

9939 

*0141 

*0342 

*0543 

*0744 

5.0 

1.6 0944 

1144 

1343 

1542 

1741 

1939 

2137 

2334 

2531 

2728 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


92 































































































































VIII 


FIVE-PLACE NATURAL LOGARITHMS 


93 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.0 

1.6 0944 

1144 

1343 

1542 

1741 

1939 

2137 

2334 

2531 

2728 

5.1 

2924 

3120 

3315 

3511 

3705 

3900 

4094 

4287 

4481 

4673 

5.2 

4866 

5058 

5250 

5441 

5632 

5S23 

6013 

6203 

6393 

6582 

5.3 

6771 

6959 

7147 

7335 

7523 

7710 

7896 

8083 

8269 

8455 

5.4 

8640 

8825 

9010 

9194 

9378 

9562 

9745 

9928 

*0111 

*0293 

5.5 

1.7 0475 

0656 

0838 

1019 

1199 

1380 

1560 

1740 

1919 

2098 

5.6 

2277 

2455 

2633 

2811 

2988 

3166 

3342 

3519 

3695 

3871 

5.7 

4047 

4222 

4397 

4572 

4746 

4920 

5094 

5267 

5440 

5613 

5.8 

5786 

5958 

6130 

6302 

6473 

6644 

6815 

6985 

7156 

7326 

5.9 

7495 

7665 

7834 

8002 

8171 

8339 

8507 

8675 

8842 

9009 

6.0 

9176 

9342 

9509 

9675 

9840 ' 

*0006 

*0171 

*0336 

*0500 

*0665 

6.1 

1.8 0829 

0993 

1156 

1319 

1482 

1645 

1808 

1970 

2132 

2294 

6.2 

2455 

2616 

2777 

2938 

3098 

3258 

3418 

3578 

3737 

3896 

6.3 

4055 

4214 

4372 

4530 

4688 

4845 

5003 

5160 

5317 

5473 

6.4 

5630 

5786 

5942 

6097 

6253 

6408 

6563 

6718 

6872 

7026 

6.5 

7180 

7334 

7487 

7641 

7794 

7947 

8099 

8251 

8403 

8555 

6.6 

8707 

8858 

9010 

9160 

9311 

9462 

9612 

9762 

9912 

*0061 

6.7 

1.9 0211 

0360 

0509 

0658 

0806 

0954 

1102 

1250 

1398 

1545 

6.8 

1692 

1839 

1986 

2132 

2279 

2425 

2571 

2716 

2862 

3007 

6.9 

3152 

3297 

3442 

3586 

3730 

3874 

4018 

4162 

4305 

4448 

7.0 

4591 

4734 

4876 

5019 

5161 

5303 

5445 

5586 

5727 

5869 

7.1 

6009 

6150 

6291 

6431 

6571 

6711 

6851 

6991 

7130 

7269 

7.2 

7408 

7547 

7685 

7824 

7962 

8100 

8238 

8376 

8513 

8650 

7.3 

8787 

8924 

9061 

9198 

9334 

9470 

9606 

9742 

9877 

*0013 

7.4 

2.0 0148 

0283 

0418 

0553 

068 ? 

0821 

0956 

1089 

1223 

1357 

7.5 

1490 

1624 

1757 

1890 

2022 

2155 

2287 

2419 

2551 

2683 

7.6 

2815 

2946 

3078 

3209 

3340 

3471 

3601 

3732 

3862 

3992 

7.7 

4122 

4252 

4381 

4511 

4640 

4769 

4898 

5027 

5156 

5284 

7.8 

5412 

5540 

5668 

5796 

5924 

6051 

6179 

6306 

6433 

6560 

7.9 

6686 

6813 

6939 

7065 

7191 

7317 

7443 

7568 

7694 

7819 

8.0 

7944 

8069 

8194 

8318 

8443 

8567 

8691 

8815 

8939 

9063 

8.1 

9186 

9310 

9433 

9556 

9679 

9802 

9924 

*0047 

*0169 

*0291 

8.2 

2.1 0413 

0535 

0657 

0779 

0900 

1021 

1142 

1263 

1384 

1505 

8.3 

1626 

1746 

1866 

1986 

2106 

2226 

2346 

2465 

2585 

2704 

8.4 

2823 

2942 

3061 

3180 

3298 

3417 

3535 

3653 

3771 

3889 

8.5 

4007 

4124 

4242 

4359 

4476 

4593 

4710 

4827 

4943 

5060 

8.6 

5176 

5292 

5409 

5524 

5640 

5756 

5871 

5987 

6102 

6217 

8.7 

6332 

6447 

6562 

6677 

6791 

6905 

7020 

7134 

7248 

7361 

8.8 

7475 

7589 

7702 

7816 

7929 

8042 

8155 

8267 

8380 

8493 

8.9 

8605 

8717 

8830 

8942 

9054 

9165 

9277 

9389 

9500 

9611 

9.0 

9722 

9834 

9944 

*0055 

*0166 

*0276 

*0387 

*0497 

*0607 

*0717 

9.1 

2.2 0827 

0937 

1047 

1157 

1266 

1375 

1485 

1594 

1703 

1812 

9.2 

1920 

2029 

2138 

2246 

2354 

2462 

2570 

2678 

2786 

2894 

9.3 

3001 

3109 

3216 

3324 

3431 

3538 

3645 

3751 

3858 

3965 

9.4 

4071 

4177 

4284 

4390 

4496 

4601 

4707 

4813 

4918 

5024 

9.5 

5129 

5234 

5339 

5444 

5549 

5654 

5759 

5863 

5968 

6072 

9.6 

6176 

6280 

6384 

6488 

6592 

6696 

6799 

6903 

7006 

7109 

9.7 

7213 

7316 

7419 

7521 

7624 

7727 

7829 

7932 

8034 

8136 

9 8 

8238 

8340 

8442 

8544 

8646 

8747 

8849 

8950 

9051 

91 o 2 i 

9.9 

9253 

9354 

9455 

9556 

9657 

9757 

9858 

9958 

*0058 

*0158 

10.0 

2.3 0259 

0358 

0458 

0558 

0658 

0757 

0857 

0956 

1055 

1154 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



































































































































































94 


CONSTANTS WITH THEIR LOGARITHMS 


IX 


Base of Naperian logarithms. 

Modulus of common logarithms .... 

Reciprocal of modulus. 

Circumference of a circle in degrees . . • 

Circumference cf a circle in minutes . 
Circumference of a circle in seconds . 

Radian expressed in degrees. 

Radian expressed in minutes. 

Radian expressed in seconds. 

Ratio of a circumference to diameter . 

7T = 3.14159 26535 89793 23846 26433 8328 


Number 

e = 2.71828183 
u = 0.43429448 

- = 2.30258509 
u 

= 360 
= 21600 
= 1296000 
= 57.29578 
= 3437.7468 
= 206264.806 
x = 3.14159265 


Number 

2x = 6.28318531 
4ir = 12.56637061 

- = 1.57079633 
2 

- = 1.04719755 

3 

— = 4.18879020 
3 

- = 0.78539816 

4 

^ = 0.52359878 
6 

- = 0.31830989 

X 

— = 0.15915494 
2ir 

- = 0.95492966 

‘,r 

- = 1.27323954 

X 


Logarithm 

0.7981799 

1.0992099 

0.1961199 

0.0200286 

V 

0.6220886 

9.8950899-10 

9.7189986-10 

9.5028501-10 

9.2018201-10 

9.9799714-10 

0.1049101 


r 2 = 


Vx 

1 

Vx 

fa 


>/!- 


9.86960440 

0.10132118 

1.77245385 

0.56418958 

0.97720502 


» -7T 


1 

Vi 

3 


a/S 


1.12837917 
= 1.46459189 
= 0.68278406 
= 2.14502940 
= 0.62035049 
= 0.80599598 


If the radius r = 1, the length of the arc is 


for 1 degree 
for 1 minute 
for 1 second 


x 

180 

x 

10800 

x 


648000 
sin 1" 


= 0.01745329 

= 0.00029089 

= 0.00000485 
- 0.00000485 


Logarithm 

0.4342945 

9.6377843-10 

2.5563025 

4.3344538 

6.1126050 

1.7581226 

3.5362739 

5.3144251 

0.4971499 

0.9942997 

9.0057003-10 

0.2485749 

9.7514251-10 

9.9899857-10 

0.0524551 

0.1657166 

9.8342834-10 

0.3314332 

9.7926371-10 

9.9063329-10 

8.2418774-10 

6.4637261-10 

4.6855749-10 

4.6855749-10 
























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